Talk:Level (logarithmic quantity)

The octave is a unit of frequency level
The reason the octave is relevant to this article is that it is unit of a logarithmic quantity. Specifically it is the unit of frequency level when the base of the logarithmic is 2, i.e. L_f = lb(f/f0) oct, where f is the frequency, f0 is a reference frequency. Dondervogel 2 (talk) 15:25, 19 November 2014 (UTC)
 * Are you sure? I've never heard of level being applied to things like frequency.  Dicklyon (talk) 15:47, 19 November 2014 (UTC)
 * ANSI S1.1-2013 Acoustical Terminology contains the following definition of frequency level

3.05 frequency level. Logarithm of the ratio of a given frequency to an appropriate reference value. The base of the logarithm and reference value should be indicated. NOTE 1 If the base of the logarithm is 2, the unit of frequency level is the octave. The reference value is approximately equal to 16.352 Hz for musical acoustics. NOTE 2 If the base of the logarithm is 21/12, the unit of frequency level is the semitone. The reference value is approximately equal to 16.352 Hz for musical acoustics (see 12.23). NOTE 3 If the base of the logarithm is 10, then the unit of frequency level is the decade; e.g., 20,000 Hz is three decades above 20 Hz.
 * Dondervogel 2 (talk) 21:13, 19 November 2014 (UTC)
 * Living, learning! Could you please add this info in octave, preferably linking to here. Then please revert my revert. Thanks! Fgnievinski (talk) 01:36, 20 November 2014 (UTC)
 * Fascinating. That makes sense that if it was used anywhere, it would be by the same standards guys who brought us the funny definition of level.  I bet you don't find anyone using it.  Dicklyon (talk) 02:26, 20 November 2014 (UTC)
 * I agree it's not used often, but here are some examples. I don't have time to do any editing today, nor for the rest of November. I might come back to it in December.  Dondervogel 2 (talk) 07:26, 20 November 2014 (UTC)
 * I found a moment to add octave and semitone to the article as additional examples of units level. Dondervogel 2 (talk) 22:05, 20 November 2014 (UTC)
 * That's pretty interesting that "frequency level" goes back to Fletcher 1934. Good find.  It doesn't look like it's used enough for engineers to know about it though. Dicklyon (talk) 00:58, 2 January 2018 (UTC)

"root-power quantity" vs. "field quantity"
Presently the sections and  are identical, but for the replacement of the term and an indication that a root-power quantity is a field quantity, but not necessarily the other way around. The linked Field, power, and root-power quantities sheds no light on the distinction (even before my recent edits). This is assured to confuse. So perhaps we should try to find what is meant in each context.

ISO 80000-3:2007 says "Since a field quantity is defined as a quantity whose square is proportional to power when it acts on a linear system, a square root is introduced in the expression of the level of a power quantity."

ISO 80000-1:2009 says "A root-power quantity is a quantity, the square of which is proportional to power when it acts on a linear system. Earlier such quantities have been called field quantities in connection with logarithmic quantities, but this name is now deprecated because field quantity has another meaning, i.e. a quantity depending on position vector r."

The second quote goes to pains to point out that the meaning of "field quantity" in this context is unrelated to the "normal" meaning ("because field quantity has another meaning"). The clause "Earlier such quantities have been called field quantities" is normal English usage to imply an equivalence of meaning (though the language technically might imply inclusion, not equivalence). The wordings of the definitions of the two terms are exactly equivalent :
 * "a quantity whose square is proportional to power when it acts on a linear system"
 * "a quantity, the square of which is proportional to power when it acts on a linear system"

I would be interested in any argument that shows that the meanings of terms are not exactly equivalent (but for deprecation of the one). At a meta-level, it is evident to me that the people who drafted the standard considered them to be exactly equivalent in meaning as used. (As an aside, I realize that ISO 80000 may not be the only source, but I will still argue that in the context of levels, any meaning of "field quantity" is inherently is dependent on a time-diffused, i.e. non-instantaneous, quantity, and hence that no sensible definition could be broader than the definition of a root-power quantity .) —Quondum 15:33, 25 December 2017 (UTC)
 * The question "what's the difference between a field quantity and a root-power quantity?" is a good one. My understanding is that rms voltage is both a field quantity and a root-power quantity, whereas voltage (instantaneous value or any statistic thereof other than rms is a field quantity but not a root-power quantity). Is my understanding correct? I'm not sure.  If we can find an RS that states they are one and the same we should state that explicitly.  If we cannot, who are we to make the inference? Dondervogel 2 (talk) 20:38, 25 December 2017 (UTC)


 * I think that I've shown that your understanding is incompatible with the wording in ISO 80000, and that my edit is justified. And in any event, I see no justification in separately listing the two terms.  I'm intrigued that do you not find the effectively identical wording in the definitions compelling.  —Quondum 04:47, 26 December 2017 (UTC)
 * Don't get me wrong. Taken in isolation I *do* find your cited text compelling.  I am conscious of not yet having provided compelling arguments to support my statement that "field quantity" and "root-power quantity" are  not synonyms.  My concern is that I have obtained a different impression by reading different standards and (possibly) different parts of these same standards.  As a gesture of good faith I shall revert my own revert, not because I agree with you but because it seems unreasonable to insist on a position I have not (yet) been able to support with evidence from RS. Dondervogel 2 (talk) 15:33, 26 December 2017 (UTC)
 * @Quondum After a careful read of ISO 80000-3:2006, I am now finding myself agreeing with you that the root-power quantity of ISO 80000-1 is entirely equivalent to (and replaces?) the field quantity of ISO 80000-3. I previously had the impression that a field quantity could be complex while a root-power quantity was necessarily real, but I now see that both are in general complex quantities. I therefore retract my previous statements and apologise for reverting your edit.  My only remaining concern is that the complex nature of the field/root-power quantity (and by implication of their levels) is not presently apparent in the article. Dondervogel 2 (talk) 23:32, 26 December 2017 (UTC)
 * np – I enjoy your careful rigour.  I had a similar concern: the complex logarithm is mentioned in ISO 80000-1:2009, but there is also a strong connection made in ISO 80000:3-2007 between Np and rad as respectively the units of the real part (the level) and imaginary part (the phase) of a single coherent (in both senses) complex quantity, in a context where there is no time-averaging, so I'm striking a my contrary claim above.  I would like to include the essence of this complex-number aspect, though wording will need thought.  With reference to your "the complex nature of the field/root-power quantity (and by implication of their levels)", my interpretation is that the level itself is not considered to be a complex quantity, only that it is the real part of a complex quantity.  The complex quantity seems to live in the s-plane, for which a suitable name is elusive ("complex frequency domain parameter"?).  This is all complicated by that the standards clearly lack underlying rigour; e.g., the radian as a unit of phase angle (an inherently scalar quantity) should not be confused with the unit of geometric angle (which is implicitly a bivector quantity) – this is like confusing units of energy and torque.  —Quondum 18:59, 27 December 2017 (UTC)
 * Once again I find myself pleading for more time to answer your question properly, but my recollection is that the level itself may be complex. My recollection/gut feeling has proven incorrect before so I'm not banking on it now, and further the answer depends on how "level" is defined, which as you point out depends on the source.  I will look it up and give you a better answer tomorrow. Dondervogel 2 (talk) 19:15, 27 December 2017 (UTC)
 * Heh – and I was implying that I'll need time too, so no conflict there. Please do not rush in any sense.  I wasn't aware that I'd implied a question, though I do appreciate learning from broader perspectives and more comprehensive information.  I would not be particularly surprised at the name of real component (e.g. "level" or "growth constant") being generalized by someone to the complex domain: it is more natural and less confusing than generalizing the name of the imaginary component (e.g. "phase" or "angular frequency"). —Quondum 19:35, 27 December 2017 (UTC)

List of relevant sources
Let's start with a list of sources. The most important ones are Also relevant are Below I've started a sub-section on each - feel free to edit these or add new ones Dondervogel 2 (talk) 09:43, 28 December 2017 (UTC)
 * ISO 80000-3:2006
 * IEC 60027-3:2002
 * ISO 80000-1:2009
 * ANSI S1.1-2013
 * I found 2 more relevant sources, both pre-dating ISO/IEC 80000:
 * Page 1974
 * Thor 1994
 * Dondervogel 2 (talk) 22:25, 7 April 2018 (UTC)
 * Those links are not working for me. Can you put them some place or email them? Dicklyon (talk) 00:28, 8 April 2018 (UTC)
 * Sorry, I got the format wrong. The links should work now. They both make fascinating reading but neither have definitive answers. Dondervogel 2 (talk) 09:41, 8 April 2018 (UTC)
 * I just added one more (Mills et al, 2001). The discussion preceding their Table 4 gets very close to the heart of the matter.  The text itself appears to refer to complex "attenuation factor" (ie, complex level difference), which I saw as supporting my interpretation, but the table itself supports your interpretation.  In authors, the authors of this article intend level to mean the real part of the complex logarithm (unit neper or bel), while phase is the imaginary part (unit radian).  Dondervogel 2 (talk) 10:07, 8 April 2018 (UTC)

ISO 80000-3:2006
Clause 0.5 Remark on logarithmic quantities and their units includes the text
 * "The taking of logarithms of complex quantities is usefully carried out only with the natural logarithm. In this International Standard, the level L_F of a field quantity F is therefore defined by convention as the natural logarithm of a ratio of the field quantity and a reference value F_0, L_F=ln(F/F_0), in accordance with decisions by CIPM and OIML. Since a field quantity is defined as a quantity whose square is proportional to power when it acts on a linear system, a square root is introduced in the expression of the level of a power quantity ... when defined by convention using the natural logarithm, in order to make the level of the power quantity equal to the level of the corresponding field quantity when the proportionality factors are the same for the considered quantities and the reference quantities, respectively. See IEC 60027-3:2002, subclause 4.2.2)" ...
 * ... and "Meaningful measures of power quantities generally require time averaging to form a mean-square value that is proportional to power. Corresponding field quantities may then be obtained as the root-mean-square value. For such applications, the decimal (base 10) logarithm is generally used to form the level of field or power quantities. However, the natural logarithm could also be used for these applications, especially when the quantities are complex."

IEC 60027-3:2002
"Complex notation is frequently used for field quantities ... The taking of logarithms of complex-quantity ratios is usefully done only with the natural logarithm ..." "A level, symbol L, is the logarithmic ratio of two field quantities or two power quantities where the quantity in the denominator is a reference quantity of the same kind as the quantity in the numerator. Complex levels are not customary. Therefore, levels are generally given in decibels. The difference of two levels determined with the same reference quantity is independent of the value of the reference quantity."
 * Section 4.1 Logarithmic ratios of field quantities includes the text (p11)
 * Section 4.3 Levels begins

ISO 80000-1:2009
Appendix C Logarithmic quantities and their units reads "C.1 General Logarithmic quantities are quantities defined by means of logarithmic functions. For a definition to be complete, the base of the logarithm must be specified. Depending on the source of the argument of the logarithm, logarithmic quantities are classified as follows: a) logarithmic ratios that are defined by the logarithm of the ratio of two quantities of the same kind; b) logarithmic quantities, in which the argument is given explicitly as a number, e.g. logarithmic informationtheory quantities; c) other logarithmic quantities. The logarithm to any specified base of an argument gives the same information about the physical situation under consideration as does the argument itself. Quantities defined with different bases are proportional to each other but have different values and are thus different quantities. In a given field of application, only logarithms of the same base shall be used. C.2 Logarithmic root-power quantities A root-power quantity is a quantity, the square of which is proportional to power when it acts on a linear system. Earlier such quantities have been called field quantities in connection with logarithmic quantities, but this name is now deprecated because field quantity has another meaning, i.e. a quantity depending on position vector r. For sinusoidal time-varying root-power quantities, the root-mean square value is the argument of the logarithm. For non-sinusoidal root-power quantities, the root-mean square value over an appropriate time interval to be specified is used. For a periodic quantity, the appropriate time interval is one cycle. Complex notation is frequently used for sinusoidal root-power quantities, for example in telecommunications and acoustics. The taking of logarithms of complex-quantity ratios is usefully done with, and only with, the natural logarithm. Many other mathematical relations and operations also become simpler if, and only if, the natural logarithm is used. That is why natural logarithms are used in the International System of Quantities, ISQ. With natural logarithms, the unit neper, symbol Np, becomes the unit coherent with the SI, but it is not yet adopted by CGPM as an SI unit. In theoretical calculations, neper, Np, for amplitude, together with radian, rad, for the phase angle, result naturally from complex notation and natural logarithms. Nonetheless the bel, symbol B, and its submultiple decibel, symbol dB, is ― for historical reasons ― very common in applications where only the amplitude, and not the phase, is considered. The bel is based on the decimal logarithm. C.3 Logarithmic power quantities A quantity that is proportional to power is called a power quantity. In many cases, energy-related quantities are also labelled as power quantities in this context. When a power quantity is proportional to the square of a corresponding root-power quantity, the numerical value of the logarithmic quantity is the same because the factor 1/2 is included in the definition of the logarithmic power quantity. C.4 Logarithmic information-theory quantities In information theory, logarithms with three different bases are used. The three bases of the logarithm are 2, e, and 10. The units of the corresponding quantities are: shannon, Sh; natural unit of information, nat; and hartley, Hart, respectively."

ANSI S1.1-2013
The ANSI definition reads "3.01 level. In acoustics, logarithm of the ratio of a variable quantity to a corresponding reference value of the same units. The base of the logarithm, the reference value, and the kind of level are to be specified." This is follows by 7 explanatory notes, none of which mention complex quantities.

Thor 1994
New International Standards for Quantities and Units Abstract: Each coherent system of units is based on a system of quantities in such a way that the equations between the numerical values expressed in coherent units have exactly the same form, including numerical factors, as the corresponding equations between the quantities. The highest international body responsible for the International System of Units (SI) is the Conférence Générale des Poids et Mesures (CGPM). However, the CGPM is not concerned with quantities or systems of quantities. That question lies within the scope of Technical Committee number twelve of the International Organization for Standardization (ISO/TC 12). Quantities, units, symbols, conversion factors. To fulfil its responsibility, ISO/TC 12 has prepared the International Standard ISO 31, Quantities and Units, which consists of fourteen parts. The new editions of the different parts of the International Standard are briefly presented here.

Mills et al 2001
Definitions of the units radian, neper, bel and decibel Abstract: The definition of coherent derived units in the International System of Units (SI) is reviewed, and the important role of the equations defining physical quantities is emphasized in obtaining coherent derived units. In the case of the dimensionless quantity plane angle, the choice between alternative definitions is considered, leading to a corresponding choice between alternative definitions of the coherent derived unit - the radian, degree or revolution. In this case the General Conference on Weights and Measures (CGPM) has chosen to adopt the definition that leads to the radian as the coherent derived unit in the SI. In the case of the quantity logarithmic decay (or gain), also sometimes called decrement, and sometimes called level, a similar choice of defining equation exists, leading to a corresponding choice for the coherent derived unit - the neper or the bel. In this case the CGPM has not yet made a choice. We argue that for the quantity logarithmic decay the most logical choice of defining equation is linked to that of the radian, and is that which leads to the neper as the corresponding coherent derived unit. This should not prevent us from using the bel and decibel as units of logarithmic decay. However, it is an important part of the SI to establish in a formal sense the equations defining physical quantities, and the corresponding coherent derived units

Summary

 * 1) The ANSI definition seems to assume levels are real quantities.
 * 2) IEC 60027-3:2002 states explicitly that "[c]omplex levels are not customary".
 * 3) ISO 80000-3:2006 strongly implies that the level L_F is complex if F is complex.
 * 4) ISO 80000-1:2009 states that "[t]he taking of logarithms of complex-quantity ratios is usefully done with, and only with, the natural logarithm", implying that levels can be complex

I dispute summary point 4, based on a too-short quote; more completely, it says: Here nepers are the unit of level, the real part of the complex logarithm; level is not complex (could be, as Dondervogel shows, but it's not customary, as IEC 60027-3:2002 states states). Besides, the power and root-power formulae will often disagree in their imaginary parts if we allow them to be complex, as the squaring introduces phase aliasing; so half of what the standards say is unapplicable if we allow these ratios to be complex. That doesn't diminish the value of using natural logs on ratios of complex field quantities, but it does suggest that root-power quantities and power quantity should generally be taken as real to avoid contradicitons elsewhere. Dicklyon (talk) 23:26, 6 April 2018 (UTC)
 * "Complex notation is frequently used for sinusoidal root-power quantities, for example in telecommunications and acoustics. The taking of logarithms of complex-quantity ratios is usefully done with, and only with, the natural logarithm. Many other mathematical relations and operations also become simpler if, and only if, the natural logarithm is used. That is why natural logarithms are used in the International System of Quantities, ISQ. With natural logarithms, the unit neper, symbol Np, becomes the unit coherent with the SI, but it is not yet adopted by CGPM as an SI unit. In theoretical calculations, neper, Np, for amplitude, together with radian, rad, for the phase angle, result naturally from complex notation and natural logarithms.

Discussion
My interpretation is that levels in the ISQ are in general complex, if the field quantity is complex (in fact, if you think about it, if F is complex, then ln(F/F_0) is in general also complex, so levels in the ISQ must be complex [ well ... unless F_0 is also complex, but that would be daft, wouldn't it? ]), while levels outside the ISQ (ANSI, IEC) are generally real. Dondervogel 2 (talk) 12:50, 28 December 2017 (UTC)
 * ISO 80000 seems to me to be flipping between different concepts in an undisciplined fashion, or at the very least without communicating adequately what it means. In the WP context, I regard this as insufficient.  Some thinking might not harm, though.  It seems clear that we can have complex root-power quantities (e.g. V = V0 exp &omega;t and I = I0 exp &omega;t, with V0 and I0 complex) and complex power quantities (e.g. P = VI∗).  Unfortunately, the "the square of which is proportional to power" does not work here due to the complex conjugate.  So let's substitute "the square of the absolute value of which is proportional to power", and not only do we have Np and rad as associated units in a complex logarithmic value, we have W and var as associated units for complex power.  With the frequency response of a filter, one can obtain a Bode plot, which is essentially plotting a frequency-dependent "complex level", but even here I do not see the term being used.  We'd need a text on the subject.  —Quondum 14:27, 28 December 2017 (UTC)
 * By a "text" do you mean a secondary source? Dondervogel 2 (talk) 14:47, 28 December 2017 (UTC)
 * That would be ideal, yes, though any believable literature that has a more in-depth discussion would be nice. At the moment we are sleuthing, trying to find what interpretations of terminology that are consistent with the statements.  The theory itself is straightforward.  —Quondum 01:32, 29 December 2017 (UTC)
 * I don't know of any source that talks about levels of complex quantities other than the ISO and IEC standards. Sources that refer to the level of a root-power quantity are also very rare (I looked yesterday and found just one),  but that does not stop us mentioning those.  To address your concern though I suggest we just state what the standards say, without interpretation. Dondervogel 2 (talk) 09:22, 29 December 2017 (UTC)
 * Yes, we should say only what the sources say. I only see mentions of "logarithms of complex-quantity ratios", without such a quantity necessarily actually being called a level.  —Quondum 12:29, 29 December 2017 (UTC)
 * From ISO 80000-3:2006 (emphasis is mine): "The taking of logarithms of complex quantities is usefully carried out only with the natural logarithm. In this International Standard, the level L_F of a field quantity F is therefore defined by convention as the natural logarithm of a ratio of the field quantity and a reference value F_0, L_F=ln(F/F_0), in accordance with decisions by CIPM and OIML." The clear implication of the word "therefore" in this sentence is that level of a field quantity is in general complex. I see no other way of interpreting it. Dondervogel 2 (talk) 13:02, 1 January 2018 (UTC)
 * This is fully consistent with the interpretation that the level is the real part of the logarithm: the part of the logarithm of the complex quantity associated with the amplitude, which is the root-power (field) quantity of the definition, which is −&delta;t in the first formula and −&alpha;x in the second formula. Your interpretation has certain jarring implications, which their failure to mention in their exposition implies (in my mind) that they did not consider a level as the quantities –&delta; + i&omega;t or −&gamma;x.  If these were to be treated as a single quantity such as a level, it would be appropriate to assign a single unit, namely Np, and point out the (unexpected to most) equality Np = rad, and not merely call the units Np and rad coherent with SI.  In contrast, they first break the logarithm into is (real and imaginary) component parts, and then they assign different units to each part (Np and rad).  This separation is further underscored by their use of minus signs in the expressions in the exponent.  Their argument shows quite adequately that these are the only units that are coherent with this treatment, especially when one considers these parts of the logarithm separately: degrees, turns or any familiar units of phase angle other than radians fail horribly to be coherent with any units of level other than nepers, and conversely with hartley and bel.  Had they been thinking in terms of levels as complex, I would expect them to say something like e$(&alpha; + i&beta;)⋅Np$ is the same thing as e$(&alpha;⋅Np + i&beta;⋅rad)$.  To encapsulate: the use of "therefore" implies only that the level is part of the logarithm.  —Quondum 15:04, 1 January 2018 (UTC)


 * No. It says the level L_F of a field quantity F is ... defined ... as the natural logarithm of a ratio of the field quantity and a reference value F_0, L_F=ln(F/F_0). That statement is not consistent with the level being the logarithm of the real part of F. It only makes sense if it is the logarithm of the complex quantity F. Dondervogel 2 (talk) 15:53, 1 January 2018 (UTC)


 * I must still disagree with your inference. It would be circular logic to assume that F or F$0$ are complex.  (And, this contradicts the definition as its square being proportional to a power quantity.)  And we are talking above about the real part of the logarithm of a complex value, not the logarithm of the real part of a complex value.  —Quondum 16:12, 1 January 2018 (UTC)
 * You make 3 points and I disagree with all 3 of them
 * That F is complex is made explicit elsewhere in the standard
 * F being complex does not contradict its square being proportional to power - it just implies a complex constant of proportionality
 * And the level is equal to the logarithm and not the real part of the logarithm - that too is explicit.
 * I am now convinced that the authors of ISO 80000-3:2006 intend the level of a field quantity to be complex, but I see I have not convinced you, so I guess we must agree to disagree :-0
 * On the other hand there must be things we can agree on (for example that F is complex), so perhaps we should focus on those? Dondervogel 2 (talk) 19:09, 1 January 2018 (UTC)
 * My impression from the quotes from the standards is that if one takes the log of ratio of complex amplitudes, then the level is the real part and the phase is the imaginary part (of the natural log). This is the only interpretation consistent with what level has always meant; I see no example anywhere suggesting that a level can be complex.  Furthermore, the "square of field quantity" clearly means the square of its magnitude if it is complex. Dicklyon (talk) 19:35, 1 January 2018 (UTC)
 * And I think we need to look to secondary sources, not try to interpret the intent of the committees that generated the standards docs. This 1982 CCIR report says "It should be observed that, as a result of some calculations on complex quantities, a real part in nepers and an imaginary part in radians are obtained." But you might have to go back to the search results page to see the snippet. Dicklyon (talk) 19:44, 1 January 2018 (UTC)
 * The trouble is we don't have any secondary sources, so we were discussing how to present the statements made by the relevant standards. Do we all agree that according to the standard the field quantity F can be complex? Dondervogel 2 (talk) 19:52, 1 January 2018 (UTC)
 * If the field quantity is a complex number describing the amplitude and phase of a sinusoid, then its level depends only on its magnitude, that is, the real part of the complex log if you'd rather do it that way. I don't see any other plausible way to interpret the standards and other sources. Dicklyon (talk) 20:37, 1 January 2018 (UTC)

Dondervogel, I happily agree to disagree, but that has the implication that we must refrain from making any assertion in the article on the matter (which I am happy with). I also agree with Dicklyon's assertion that we need a secondary source here, and again lack thereof forces our silence on the matter. —Quondum 02:09, 2 January 2018 (UTC)
 * Well, we have at least one source that says the log of a complex field-type quantity has a real parts that's a level and an imaginary part that's a phase. I don't see any source suggesting that a level can be complex.  So can we say what the CCIR 1982 clause says, that "a real part in nepers and an imaginary part in radians are obtained"? Dicklyon (talk) 05:24, 2 January 2018 (UTC)
 * I agree that the real part of ln(F/F0) would be in nepers and the imaginary part is in radian. Making that would make clear that ln(F/F0) is a complex quantity. Dondervogel 2 (talk)
 * The square of A$2$ of a complex amplitude A does not satisfy any reasonable definition of a power quantity (it changes wildly just by changing your time reference). This technically means that the complex amplitude A fails the definition of a field quantity.  I think we can say that $|A|$ meets the definition of a field quantity, and hence that Re(ln(A/A$0$)) is a level, but that is about the limit of what we can say.  —Quondum 12:13, 2 January 2018 (UTC)
 * I don't understand your objection. The IEC and ISO standards both state explicitly that the field quantity is complex, and carry out complex arithmetic on that complex quantity, including logarithms. Whether you or I consider it a sensible definition is irrelevant, and I fail to see the problem with acknowledging this demonstrable fact in the article. The only thing that is not there in black and white is a statement that these complex logarithms are also levels. Dondervogel 2 (talk) 15:07, 2 January 2018 (UTC)
 * afterthought: I think we would reach consensus more quickly on level if we first had consensus on the meaning of field quantity (or root-power quantity). I suggest we work on this at Field, power, and root-power quantities and return to level when we're done there. Dondervogel 2 (talk)
 * From my perspective, you are reading something into the standards that is not there. They refer to complex quantities and to field quantities, but I do not see them saying what you say is explicitly stated.  I concur that a discussion at Field, power, and root-power quantities before considering levels would be better.  —Quondum 18:08, 2 January 2018 (UTC)
 * But it is there, in black and white. For example, on page 13 of IEC 60027-3:2002 one can find the example Q_U = ln(U1/U2) = 2,303 Np + j 0,524 rad, where U1 and U2 are complex voltages equal to 30 exp(j pi/2)V and 3 exp(j pi/3)V, respectively. The only interpretation on my part (which follows directly from the definition, so it is but a small step) is to infer that Q_U is a level. The complex field quantities U1 and U2 are explicit. Dondervogel 2 (talk) 20:48, 2 January 2018 (UTC)
 * I think a more plausible interpretation is that 2,303 Np is the level and 0,524 rad is the phase shift.  They do not say (2,303 + j 0,524) Np, which they would if they intended a complex level in nepers. Dicklyon (talk) 21:25, 2 January 2018 (UTC)
 * You might consider that to be plausible, but your interpretation is in direct contradiction with the IEC/ISO definition of level. Dondervogel 2 (talk) 21:28, 2 January 2018 (UTC)
 * I can see that it's a contradiction of your interpretation of the definition, but I can't see any source that suggests that anyone agrees with your interpretation. Dicklyon (talk) 21:33, 2 January 2018 (UTC)
 * There's no interpretation in stating that the level L_F of a field quantity F is defined in the ISQ as ln(F/F_0). That is the ISQ definition.  You are arguing that the level should be defined as ln(|F|/F_0), and some standards do indeed define it in that way.  Just not the ISQ. Dondervogel 2 (talk) 21:41, 2 January 2018 (UTC)
 * I see nothing in ISO 80000 to suggest that F is implied to be complex-valued, something that is crucial to your argument. Nowhere do they say that the complex quantities they refer to are field quantities, or that the complex logarithm they use is used to obtain a level.  I see their reference to complex values solely as a rationale for why Np is a coherent unit.  (I have yet to obtain IEC 60027.)  —Quondum 23:08, 2 January 2018 (UTC)
 * You might also want to consider WP:WEIGHT. —Quondum 00:36, 3 January 2018 (UTC)
 * There's more evidence out there to support my assertion that field quantities may be complex than to support yours that the level of a root-power quantity is identical to the level of a field quantity. After careful reading of ISO 80000, I concluded that your assertion was the only logical interpretation of ISO 80000-1:2009.  Now, after careful reading of ISO 80000 and IEC 60027 I conclude that my assertion is the only logical interpretation of those documents, AND there are multiple secondary sources to support my assertion, compared with only one (that I could find) supporting yours. Dondervogel 2 (talk) 08:40, 3 January 2018 (UTC)
 * I no longer see any point in debating this. —Quondum 13:12, 3 January 2018 (UTC)

Discussion, continued
Hi. I come here from Talk:Field,_power,_and_root-power_quantities. Is power restricted to the real domain in the standards? If so, then the definition "root-power quantity is a quantity, the square of which is proportional to power" restricts root-power to the real domain, too. A complex-valued root-power (for a real-valued power) would require a different definition: "root-power quantity is a quantity, whose product against its complex conjugate is proportional to power". fgnievinski (talk) 03:13, 9 April 2018 (UTC)

My guess about the whole motivation for introducing root-power was to restrict it to real-valued numbers. The equation to keep in mind is RP ~ sqrt(P): root-power is proportional to the square-root of power. If you start with a field quantity, possibly complex-valued, then first you must obtain its power view complex conjugation, P = V^*V, only then to obtain root-power. They avoided calling it amplitude because it's normally associated with periodic waves, which would be too restrictive -- they'd have to call it "envelope" in general. And I guess that magnitude, modulus and absolute value were too generic, so they had to coin a new name. fgnievinski (talk) 03:13, 9 April 2018 (UTC)
 * I think you're right. I don't see any place suggesting that power can be anything but nonnegative real, and root-power seems intended to be also.  But the confusion is evident in the places where the standards mention complex values (barely), as pointed out above (e.g. in 80000-1, "Complex notation is frequently used for sinusoidal root-power quantities"). Dicklyon (talk) 03:26, 9 April 2018 (UTC)

Part of the problem with this discussion is that it is almost entirely based around standards documents. Remember, standards are primary sources and we should be basing articles on secondary sources. The standards themselves cannot tell you how people are actually using the standards, how widespread that use is, and if something else entirely is normal practice in a field. Secondary sources are needed for that kind of analysis. This is not an article about an international standard. It is about level as it is used in sound/electrical engineering. SpinningSpark 10:40, 9 April 2018 (UTC)
 * That's a very good point. And I think that if you ask engineers about level, you will have a hard time finding one that ever interprets it in even approximately the way the standards do.  E.g. engineers will talk about an amplifier with a gain of 20 dB (and phase delay in degrees, perhaps), but would never consider that 20 dB to be a level, or interpret that dB unit as measuring level in that context.  Level, to an engineer, is more absolute, a power or amplitude measurement, even though it's expressed as a log with respect to a reference.  In my recent book, I have a box On "Level", to discuss this in the context of automatic gain control.  Here I quote it in full:

"The concept of level, frequently found as intensity level or loudness level, is usually expressed on a logarithmic scale, in decibels. In the 1960s, standards organizations actually began to define level to be the logarithm of the ratio of an intensity to a reference intensity, so that they could cast the decibel as a unit of level, making the dB behave more like a conventional unit than as a logarithm; for example, ANSI (1960) defines level: “In acoustics, the level of a quantity is the logarithm of the ratio of that quantity to a reference quantity of the same kind. The base of the logarithm, the reference quantity, and the kind of level must be specified.” Most engineers have not been taught this definition of level, though, and use level more informally as a general notion of a measurement of how big a signal is, whether they represent it logarithmically or not. In an automatic gain control loop, we typically feed back some measurement of output level to control the system gain. Some treatments in the literature assume that output level is measured logarithmically, but this model is difficult to get to work right at very low signal levels, so is more often avoided. Wheeler (1928) speaks of “maintaining the desired signal level in the detector or rectifier,” which is much like how we treat it here. That is, we let the detection nonlinearity (the rectifier) provide a signal that we take to represent level, with no prejudice about whether it is proportional to power, or amplitude, or log power, or something else. In a real AGC system with signals representing sounds, level is a derived quantity, or even an abstraction, of what the system adapts to. A detector or rectifier produces a derived signal whose short-time average can be taken as level. But the rectified signal—whether positive part or absolute value—also contains fine temporal structure that is not part of what we call level. There may be no clean separation between the frequencies or time scales of level fluctuations and the frequencies or time scales of fine structure. But we can pretend."


 * This is on p.203 of the printed book, p.193 of the free PDF.
 * I think engineers are more likely to think in terms of the dictionary definition:


 * Look at these books that discuss decibel and level and logarithm, for example. Do you see any that even introduce or define the concept of level as a logarithm? Dicklyon (talk) 14:38, 9 April 2018 (UTC)
 * Oh no, that's a rabbit hole -- "level" as synonym of "quantity, amount, extent, measure, degree, volume, size, magnitude, intensity, proportion" is too wide open and informal. I don't think we should cover the layman's usage of the term. fgnievinski (talk) 15:12, 9 April 2018 (UTC)
 * I'm not suggesting we treat it that way, but it may be useful background for us to understand that most engineers and authors don't actually know that it has a more formal meaning than that. Dicklyon (talk) 23:00, 9 April 2018 (UTC)
 * But following up on Spinningspark's suggestion, most secondary sources will restrict level to the real domain. I challenge anyone to source a book showing otherwise. fgnievinski (talk) 15:12, 9 April 2018 (UTC)
 * A google scholar search for level, "field quantity" and "root-power quantity" finds 3 hits, all of which seem to treat "level" as a real quantity. But field quantity is often complex. Dondervogel 2 (talk) 16:47, 9 April 2018 (UTC)
 * Even though a field quantity (in one sense) may be complex, this does not mean that a level (which is the logarithm of a time-averaged window of power) is taken as a logarithm of that instantaneous or local quantity. Keep in mind that "field quantity" can mean "a function of space and time" (which would fit with being complex) rather than a root-power quantity.  Isn't this precisely the confusion that ISO 80000-1 was trying to avoid by preferring the term "root-power quantity" ("Earlier such quantities have been called field quantities in connection with logarithmic quantities, but this name is now deprecated because field quantity has another meaning, i.e. a quantity depending on position vector r.")?  —Quondum 16:59, 9 April 2018 (UTC)
 * I agree the term "field quantity" has multiple meanings, one of which is that of ISO 80000-3, for which complex notation is common-place (actually I would say it is the norm). I agree the authors of ISO 80000-1 intended to give this quantity a different name by calling it "root-power quantity". Whether in doing so they also intended to restrict it to non-negative quantities is hard to judge because the standards are silent on this point. Dondervogel 2 (talk) 18:43, 9 April 2018 (UTC)
 * All that I was pointing out is that the frequency of the term "complex field quantity" in the literature does not seem to be helpful in this context. —Quondum 21:39, 9 April 2018 (UTC)
 * That could be (I did not check the relevance of the individual hits), but even supposing all those "hits" are actually misses would does not alter my main point that physicists almost universally use complex field variables to represent acoustic and electromagnetic fields (the real part of which represents the physical quantity). Any physics text book describing solutions to the linear wave equation, Helmholtz equation or similar will demonstrate this. Dondervogel 2 (talk) 22:38, 9 April 2018 (UTC)
 * I think you're confusing two very different uses of complex values in physics and engineering. The use that's applicable here is for sinusoidal analysis (Fourier components) where the magnitude of the complex value is proportional to root power and the phase is, well, phase.  That's very different from uses where the real part represents the physical quantity, which is really just a sort of informal shorthand.  Keep in mind that the real part operator is not a linear operator.  My book has a section on that, too: Section 8.7 Keeping It Real.  Dicklyon (talk) 23:09, 9 April 2018 (UTC)
 * Also, Dondervogel's hits for "complex field quantity" are mostly not in the context of levels, and in many cases the complex values are really just 2D space vectors. So most of those hits don't connect well here. Dicklyon (talk) 05:45, 10 April 2018 (UTC)
 * Dondervogel, of your 3 book hits, consider their definitions.
 * A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound has a real definition: "According to ISO 80000-1:2009 ‘Quantities and Units Part 1: General’ (ISO, 2009), and ANSI S1.1-2013 ‘Acoustical Terminology’ (ANSI, 2013), a level, L, is the logarithm of the ratio of a quantity q to a reference value of that quantity q0. In equation form, L = logr q/q0, from which it is clear that the value of q (the nature of which must also be specified) can only be recovered unambiguously from that of L if the base of the logarithm (r) and the reference value (q0) are both known precisely." ... They go on to restrict to field quanities that are square roots of positive powers, but don't say if the standards compel that: "For every real, positive power quantity P there exists a field quantity F = P1/2, in which case that field quantity may be referred to as a root-power quantity (ISO, 2009), and for which (assuming also that F0 = P01/2) the level LF as defined above is equal to the level LP. Further, the term “field quantity” is deprecated by ISO 80000-1:2009. For these reasons, attention is restricted in the following to real, positive power quantities and to their corresponding root-power quantities."
 * UNH DRAFT Soundscape and Modeling Metadata Standard Version 2 has a definition of level that is not a very clear or careful one (or perhaps not even a definition?). It says: "In general, a level is a logarithm of a ratio of two like quantities.  A widely used level in acoustics is the level of a power quantity (ISO 80000‐3:2006; IEC 60027‐3:2002).  A power quantity is one that is proportional to power.  The level of a power quantity, P, is the logarithm of that power quantity to a reference value of the same quantity, ..." I think this is as close as most books get to defining "level" for an engineering audience; it's pretty pathetic; never precise. Dicklyon (talk) 23:42, 9 April 2018 (UTC)
 * Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis has level only in a footnote, I think: "The level of a root-power quantity LF is a logarithmic root-power quantity defined as LF = ln(F/F0) where F and F0 represent two root-power quantities of the same kind, F0 being a reference quantity;"  This tells you what "level of a root-power quantity" is, but never quite defines what a level is, nor very clearly what a root-power quantity is.  And "is a logarithmic root-power quantity" seems like quite a confusing abuse.
 * So basically, this body of knowledge hardly exists outside the standards, and even within the standards is not clear nor consistent. Most practicing engineers and physicists would be surprised, I think, to learn that "level" has a standardized definition as the logarithm of a ratio.  This notion was invented so that decibels and nepers could be treated as units (of level), but most people don't know that, and don't know that they don't know that, or see what they might want to know that.  Given this sad state, we should refrain from reading more into the standards than what they state clearly and what secondary sources interpret them as saying. Dicklyon (talk) 23:42, 9 April 2018 (UTC)
 * Note: "A Century of Sonar" seems to be a magazine article (Acoustics Today). fgnievinski (talk) 04:38, 10 April 2018 (UTC)
 * Thanks, yes, I should have said 3 scholar hits, not 3 book hits. Dicklyon (talk) 05:37, 10 April 2018 (UTC)

Discussion on rewriting statement

 * My concern is that Field, power, and root-power quantities currently states "In the analysis of signals and systems using sinusoids, field quantities and root-power quantities may be complex valued. I intend to rewrite it as follows:
 * "Field quantities, such as phasors and EM fields, are generally complex valued. In the following we restrict attention to positive real valued power. Power can be obtained multiplying the field against its complex conjugate. Root-power quantities then is also real valued, defined as the positive square root of power."
 * My intention is to sidestep the issue of complex level. Current standards clearly need improvement in that regard. fgnievinski (talk) 04:38, 10 April 2018 (UTC)
 * I'd be careful there. EM fields are never complex; when they're sinusoidal, they can be described using phasors, e.g. as the real part of complex exponentials, but the fields are real (vector-valued real, typically).  Maybe it's true that "field quantities are generally complex valued", but that's not because field values themselves are complex. Dicklyon (talk) 05:35, 10 April 2018 (UTC)
 * Thanks for the correction. Here is a second try:
 * "Field quantities, such as phasors – as the analytic representation of a voltage running in an electrical wire or of an EM field propagating in 3D space –, are generally complex valued. In the following we restrict attention to positive real valued power. Power can be obtained multiplying the field quantity against its complex conjugate. Root-power quantities then is also real valued, defined as the positive square root of power."
 * fgnievinski (talk) 15:16, 17 April 2018 (UTC) I like that. Dondervogel 2 (talk) 17:05, 17 April 2018 (UTC)
 * Both the original and suggested replacement place undue emphasis on a mathematical convenience (phasor-like quantities). It an overstatement/misstatement to say that "Field quantities [...] are generally complex valued."  My preference would be to delete the statement, not replace it.  In physics, mathematics and metrology this emphasis on complex field quantities would sound strange.  —Quondum 23:49, 17 April 2018 (UTC)
 * The emphasis on the complex nature would only sound strange to a physicist if he/she was so used to using complex numbers to describe the field that it had become second nature, and there was therefore no longer a need to point out. I do not believe it would be second nature to most WP readers, which is why I think it does need to be pointed out. Dondervogel 2 (talk) 06:06, 18 April 2018 (UTC)
 * We seem to have two groups of people in this discussion: those who think everyone routinely describes fields with phasors, and those who don't. We need to be careful that we do not present something as general if it isn't.  Being an electrical engineer, I understand how ubiquitous this description can become within the discipline, but to claim that this is general would be a mistake.   For example, "as the analytic representation [...] of an EM field propagating in 3D space" is an unusable (i.e. ill-defined) description in physics, where the inertial frame might change.  Let's stick to what we can clearly source, and exclude content for which we clearly do not have consensus.  —Quondum 13:53, 18 April 2018 (UTC)
 * No. "Field quantities, such as phasors, are generally complex valued" misrepresents what field quantities are and doesn't at all clarify what a phasor is or why we sometimes use complex numbers.  It would be more correct to say "In the analysis of linear systems using sinusoidal signal or waves, field quantities are sometimes taken to be phasors, represented abstractly by complex-valued quantities." Dicklyon (talk) 15:06, 18 April 2018 (UTC)
 * Agreed, there was undue generalization. Here's a 3rd try:
 * "Field quantities may be real valued (e.g., EM field propagating in 3D space) or complex (e.g., phasors as the analytic representation of voltage traveling in a wire). Restricting attention to nonnegative real valued power, it can be obtained multiplying the field quantity against itself or, more generally, its complex conjugate. Them, root-power quantities will also be real valued, when defined as the positive square root of power."


 * The statements in parentheses are not essential and may be removed if necessary, although it'd make the text more precise at the expense of becoming less concrete. fgnievinski (talk) 16:18, 18 April 2018 (UTC)
 * I liked it before and I like it even more now. Thank you for your persistence. Dondervogel 2 (talk) 16:34, 18 April 2018 (UTC)
 * Still no, as it suggests that the real/complex distinction is somehow related to the 3D/wire distinction, and omits the very important point that the complex notation is only sensible in the analysis of linear systems (3D or 1D or lumped doesn't matter) using sinusoidal signals.  I'll try to find time later to draft an alternate proposal. Dicklyon (talk) 16:51, 18 April 2018 (UTC)
 * Waiting for my plane. Try this:
 * "Physical field quantities are real valued, but in the analysis of linear systems using sinusoidal signals (that is, in the frequency domain), complex-valued field quantities, phasors, are sometimes used. Power is always real valued, proportional to the square of a real-valued field quantity or more generally to the square of the magnitude of a complex-valued field quantity. Root-power quantities, defined as the positive square root of power quantities, are real and positive. Logarithms of complex field quantity ratios are generally complex, having a real part representing the level, or logarithm of the magnitude of the ratio, and complex part being the phase, or angle, of the ratio (for natural logarithms, the units of level and phase arrived at this way are nepers and radians, respectively)."


 * I realize it's hard to back this up exactly with sources, but I think it's about what ought to be said. If someone objects, we can trim it. Dicklyon (talk) 19:10, 18 April 2018 (UTC)


 * That works too, except that complex field quantities are not limited to the frequency domain (analytic signals are complex representations in the time domain). Dondervogel 2 (talk) 22:00, 18 April 2018 (UTC)
 * True; I had forgotten about that. So do people use level of analytic signals?  I think it makes much less sense, since they are not physical and so have no clear relation to power. Dicklyon (talk) 00:25, 19 April 2018 (UTC)
 * The analytic signal is just the real signal added to its Hilbert transform times i. This approach breaks down in general when also a function of space: along what family of worldlines to take the transform?  —Quondum 00:43, 19 April 2018 (UTC)
 * Well, if you just think of functions of time at a point in space it might be OK. But what is the power? The squared Hilbert envelope?  Or the squared real part? I can't find any papers or books that talk about the level of analytic signals specifically, but they do sometimes talk about components of modulated signals being so many dB down, in the context of analytic signals sometimes.  Can an analytic signal act on a linear system?  Seems like yes, but then what? Not sure what to say.  Dicklyon (talk) 01:30, 19 April 2018 (UTC)
 * I think it would help to agree first on a sentence limited in scope to 'field quantity', without worrying about the implications for level. Once we have that sentence (which can go in Field, power, and root-power quantities) it will make it easier to write something about power (and level). Dondervogel 2 (talk) 08:50, 19 April 2018 (UTC)
 * Perhaps so. I've been looking for sources that associate "field quantity" with "analytic signal".  The first few hits here suggest that there can be relationships between field quantities and analtyic signals, but not that a field quantity can be an analytic signal.  I suggest we drop that option and stick closer to what I said above. Dicklyon (talk) 14:59, 19 April 2018 (UTC)
 * Also keep in mind the confusing aspect that ISO 80000 points out: that "field quantity" gets used with two different meanings. In the context of these links, the predominant sense seems to be that of a tensor-valued function on a manifold.  Unless we choose to distinguish between the two senses (the second being the root-power sense), this discussion will likely be confused.  I think it would help to start by clarifying this distinction in the article – for which ISO 80000 can serve as a starting reference.  —Quondum 17:32, 19 April 2018 (UTC)

Follow-up
a few years later, here is another attempt at reaching consensus: I interpret the previous discussion as Quondum agreeing with Dicklyon, with whom I also agree. So here is a revised version of the Dicklyon's alternate proposal: "Although physical field quantities are real valued, it is often useful to analyze them in terms of complex values (e.g., phasors, analytic signals, spectral components, etc.). Power is real valued, proportional to the absolute square of the field quantity. Root-power quantities are defined as the positive square root of power quantities, and as such are non-negative real values. Levels are always real valued and are directly proportional to the logarithms of power and of root-power quantity ratios. Levels of complex field quantity ratios are more complicated. Although logarithms of a complex field quantity ratio are generally complex, only the real part represents the level, or logarithm of the absolute value of the ratio (the imaginary part is the phase, or angle, of the ratio); for natural logarithms, the units of level and phase arrived at in this way are nepers and radians, respectively." fgnievinski (talk) 23:52, 17 May 2020 (UTC)
 * 1) power quantities are real valued
 * 2) root-power quantities are real valued
 * 3) field quantities may be complex valued
 * 4) the root-power of a field quantity is obtained taking the square root of its corresponding power, hence the name
 * 5) the square of a root-power quantity equals the corresponding power
 * 6) the absolute square of a field quantity equals the corresponding power
 * 7) level quantities are real valued
 * 8) the level of a complex field quantity may be obtained in any of three different ways:
 * 9) calculating the corresponding power or absolute square then applying the power-to-level formula;
 * 10) calculating the corresponding root-power or absolute value then applying the root-power-to-level formula;
 * 11) applying the root-power-to-level formula directly to the complex field quantity then extracting the real component of the result (not to be confused with applying the root-power-to-level formula to the real part of the complex field quantity).
 * 12) no other system is internally consistent.
 * 13) root-power quantities were introduced to avoid the complications in the third way of calculating the level of a complex field quantity above.
 * It appears that ISO 80000-3:2019 makes no mention of levels, in contrast to ISO 80000-3:2006, which was our main source on this. To my memory, ISO 80000-1:2009 and ISO:80000-3:2006 were the only parts that made mention of levels, though one might expect ISO 80000-8 to do so.  It might make sense to wait until ISO 80000-1 and maybe ISO 80000-6 are published (which should be within a year, I guess) before proceeding.  We may need to completely rethink what goes into this article.  —Quondum 00:18, 18 May 2020 (UTC)
 * That will be interesting! In the mean time, I do like the way Fgnievinski put it above. Dicklyon (talk) 03:21, 18 May 2020 (UTC)
 * I don't expect either ISO 80000-1 or ISO 80000-6 to help us with this, as I expect both to avoid defining logarithmic quantities, which instead would appear a few years down the line in IEC 80000-15. My suggestion for fixing the gaping hole caused by the revision of ISO 80000-3 is to first compile a list of published sources (including secondary ones, so we don't rely so much on a single primary source) that define or use "level" and then rewrite the article based on our list. Dondervogel 2 (talk) 06:36, 18 May 2020 (UTC)
 * I think Fgnievinski's summary is pretty neat. A noteworthy point is the separation of root-power quantity from field quantity.  It does not deal with one aspect, which is the time-dependence (or averaging intervals), which can be pretty thorny with signals that "vary in amplitude over time".  Power and root-power quantities inherently are averaged over time in some sense, whereas complex field quantities seem to be instantaneous time-dependent values.  One way to deal with this seems to be by assuming that the signal is the real part of a positive-only frequency spectrum of which the power is calculated (requiring a Hilbert transform to calculate).  Without going into such detail (which would amount to OR, but then so does the above IMO), in the article we could only deal with a definition for stationary signals where the averaging interval is immaterial.  This is all made pretty difficult by there apparently not being any really consistent picture out there.  In reality, I think we're stuck with what Dondervogel2 says, which is to try to distill some kind of picture from the literature (more in the scope of a thesis than a WP article) and hope that IEC 80000-15 brings a semblance of consistent picture.  —Quondum 13:08, 18 May 2020 (UTC)
 * I started to make some lists (below), using selected results from 3 searches on Google Scholar. One reference (D'Amore 2015) found its way into all 3 lists, suggesting that might be a good place to start. Dondervogel 2 (talk) 16:47, 18 May 2020 (UTC)
 * I added a 4th list and then combined all 4. Still missing are the IEC, IEEE and ANSI standards that define some of these concepts, but I suggest we leave those to one side as they are subject to revision. Following secondary sources will probably result in a more stable article. Dondervogel 2 (talk) 08:07, 19 May 2020 (UTC)

Hits for “level of a field quantity”

 * 1) Ainslie, M. A. (2015). A century of sonar: Planetary oceanography, underwater noise monitoring, and the terminology of underwater sound. Acoustics Today, 11(1), 12-19.
 * 2) Aubrecht, G. J., French, A. P., & Iona, M. (2011). About the International System of Units (SI) Part II. Organization and General Principles. The Physics Teacher, 49(9), 540-543.
 * 3) Thompson, A., & Taylor, B. N. (2008). Use of the international system of units (SI).
 * 4) Thoreson, E. J. (2002). Apparatus to Deliver Light to the Tip-sample Interface of an Atomic Force Microscope (AFM).
 * 5) Valdés, J. (2002). The unit one, the neper, the bel and the future of the SI. Metrologia, 39(6), 543.

Hits for “level of a power quantity”

 * 1) Ainslie, M. A. (2015). A century of sonar: Planetary oceanography, underwater noise monitoring, and the terminology of underwater sound. Acoustics Today, 11(1), 12-19.
 * 2) D’Amore, F. Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis.
 * 3) Slabbekoorn, H., Dalen, J., de Haan, D., Winter, H. V., Radford, C., Ainslie, M. A., ... & Harwood, J. (2019). Population‐level consequences of seismic surveys on fishes: An interdisciplinary challenge. Fish and Fisheries, 20(4), 653-685.
 * 4) Thompson, A., & Taylor, B. N. (2008). Use of the international system of units (SI).
 * 5) Thoreson, E. J. (2002). Apparatus to Deliver Light to the Tip-sample Interface of an Atomic Force Microscope (AFM).

Hits for “level of a root-power quantity”

 * 1) D’Amore, F. Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis.

Hits for “level"+"decibel"+"neper"

 * 1) Mills, I. M. (1995). Unity as a unit. Metrologia, 31(6), 537.
 * 2) Mills, I. M., Taylor, B. N., & Thor, A. J. (2001). Definitions of the units radian, neper, bel and decibel. Metrologia, 38(4), 353.
 * 3) Mills, I., & Morfey, C. (2005). On logarithmic ratio quantities and their units. Metrologia, 42(4), 246.
 * 4) Valdés, J. (2002). The unit one, the neper, the bel and the future of the SI. Metrologia, 39(6), 543.
 * 5) Young, R. W. (1939). Terminology for logarithmic frequency units. The Journal of the Acoustical Society of America, 11(1), 134-139.

All 4 searches combined, in date order

 * 1) Young, R. W. (1939). Terminology for logarithmic frequency units. The Journal of the Acoustical Society of America, 11(1), 134-139.
 * 2) Mills, I. M. (1995). Unity as a unit. Metrologia, 31(6), 537.
 * 3) Mills, I. M., Taylor, B. N., & Thor, A. J. (2001). Definitions of the units radian, neper, bel and decibel. Metrologia, 38(4), 353.
 * 4) Valdés, J. (2002). The unit one, the neper, the bel and the future of the SI. Metrologia, 39(6), 543.
 * 5) Thoreson, E. J. (2002). Apparatus to Deliver Light to the Tip-sample Interface of an Atomic Force Microscope (AFM).
 * 6) Mills, I., & Morfey, C. (2005). On logarithmic ratio quantities and their units. Metrologia, 42(4), 246.
 * 7) Thompson, A., & Taylor, B. N. (2008). Use of the international system of units (SI).
 * 8) Aubrecht, G. J., French, A. P., & Iona, M. (2011). About the International System of Units (SI) Part II. Organization and General Principles. The Physics Teacher, 49(9), 540-543.
 * 9) Ainslie, M. A. (2015). A century of sonar: Planetary oceanography, underwater noise monitoring, and the terminology of underwater sound. Acoustics Today, 11(1), 12-19.
 * 10) D’Amore, F. (2015) Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis.
 * 11) Slabbekoorn, H., Dalen, J., de Haan, D., Winter, H. V., Radford, C., Ainslie, M. A., ... & Harwood, J. (2019). Population‐level consequences of seismic surveys on fishes: An interdisciplinary challenge. Fish and Fisheries, 20(4), 653-685.

Young (1939)
Fletcher* has proposed the use of a logarithmic frequency scale such that the frequency level equals the number of octaves, tones, or semitones that a given frequency lies above a reference frequency of 16.35 cycles/sec., a frequency which is in the neighborhood of that producing the lowest pitch audible to the average ear. The merits of such a scale are here briefly discussed, and arguments are presented in favor of this choice of reference frequency. Using frequency level as a count of octaves or semitones from the reference Co, a rational system of subscript notation follows logically for the designation of musical tones without the aid of staff notation. In addition to certain conveniences such as uniformity of characters and simplicity of subscripts (the eight C's of the piano, for example, are represented by C• to C8) this method shows by a glance at the subscript the frequency level of a given tone counted in octaves from the reference C0=16.352 cycles/sec. From middle Cl, frequency 261.63 cycles/sec., the interval is four octaves to the reference frequency, so that below Cl there are roughly four octaves of audible sound. Various subdivisions of the octave are considered in light of their ease of calculation and significance, and the semitone, including its hundredth part, the cent, is shown to be suitable. Consequently, for general use in which a unit smaller than the octave is necessary it is recommended that frequency level counted in semitones from the reference frequency be employed.
 * H. Fletcher, J. Acous. Soc. Am. 6, 59-69 (1934).

Mills (1995)
The arguments for regarding the number 1 as a unit of the SI are reviewed. Examples of dimensionless quantities are presented, and problems associated with representing the values of dimensionless quantities of very small magnitude are discussed. The logarithmic ratio units bel, decibel and neper are discussed.

Mills et al. (2001)
The definition of coherent derived units in the International System of Units (SI) is reviewed, and the important role of the equations defining physical quantities is emphasized in obtaining coherent derived units. In the case of the dimensionless quantity plane angle, the choice between alternative definitions is considered, leading to a corresponding choice between alternative definitions of the coherent derived unit - the radian, degree or revolution. In this case the General Conference on Weights and Measures (CGPM) has chosen to adopt the definition that leads to the radian as the coherent derived unit in the SI. In the case of the quantity logarithmic decay (or gain), also sometimes called decrement, and sometimes called level, a similar choice of defining equation exists, leading to a corresponding choice for the coherent derived unit - the neper or the bel. In this case the CGPM has not yet made a choice. We argue that for the quantity logarithmic decay the most logical choice of defining equation is linked to that of the radian, and is that which leads to the neper as the corresponding coherent derived unit. This should not prevent us from using the bel and decibel as units of logarithmic decay. However, it is an important part of the SI to establish in a formal sense the equations defining physical quantities, and the corresponding coherent derived units.

Valdés (2002)
The 21st Conférence Générale des Poids et Mesures (CGPM) considered in 1999 a resolution proposing that the neper rather than the bel should be adopted as the coherent derived SI unit. Discussions remain open for further considerations until the next CGPM in 2003. In this paper further arguments are presented showing the confusions generated by the use of some dimensionless units, while the changes that the SI will have to face in the future are of a quite different nature.

Thoreson (2002)
An apparatus for the delivery of radiation to the tip-sample interface of an Atomic Force Microscope (AFM) is demonstrated. The Pulsed Light Delivery System (PLDS) was fabricated to probe photoinduced conformational changes of molecules using an AFM. The PLDS is 67 mm long, 59 mm wide, and 21 mm high, leaving clearance to mount the PLDS and a microscope slide coated with a thin film of photoactive molecules beneath the cantilever tip of a stand-alone AFM. The PLDS is coupled into a fiber pigtailed Nd:Yag frequency doubled laser, operating at a wavelength of 532 nm. The radiation delivered to a sample through the PLDS can be configured for continuous or pulsed mode. The maximum continuous wave (CW) power delivered was 0.903 mW and the minimum pulse width was 12.3 ms (maximal 401 ms), corresponding to a minimal energy of 0.150 nJ (maximal 362 nJ), and had a cycle duration of 10.0 ms. The PLDS consists of micro-optical components 3.0 mm and smaller in diameter. The optical design was inspired by the three-beam pick-up method used in CD players, which could provide a method to focus the pulse of light onto the sample layer. In addition, the system can be easily modified for different operational parameters (pulse width, wavelength, and power). As proof that the prototype design works, we observed a photoinduced ‘bimetallic’ bending of the cantilever, as evidenced by observing no photoinduced bending when a reflective-coated cantilever was replaced by an uncoated cantilever. Using the apparatus will allow investigation of many different types of molecules exhibiting photoinduced isomerization.

Mills & Morfey (2005)
The use of special units for logarithmic ratio quantities is reviewed. The neper is used with a natural logarithm (logarithm to the base e) to express the logarithm of the amplitude ratio of two pure sinusoidal signals, particularly in the context of linear systems where it is desired to represent the gain or loss in amplitude of a single-frequency signal between the input and output. The bel, and its more commonly used submultiple, the decibel, are used with a decadic logarithm (logarithm to the base 10) to measure the ratio of two power-like quantities, such as a mean square signal or a mean square sound pressure in acoustics. Thus two distinctly different quantities are involved. In this review we define the quantities first, without reference to the units, as is standard practice in any system of quantities and units. We show that two different definitions of the quantity power level, or logarithmic power ratio, are possible. We show that this leads to two different interpretations for the meaning and numerical values of the units bel and decibel. We review the question of which of these alternative definitions is actually used, or is used by implication, by workers in the field. Finally, we discuss the relative advantages of the alternative definitions.

Thompson & Taylor (2008)
Introduction/Purpose of Guide The International System of Units was established in 1960 by the 11th General Conference on Weights and Measures (CGPM— see Preface). Universally abbreviated SI (from the French Le Système International d’Unités), it is the modern metric system of measurement used throughout the world. This Guide has been prepared by the National Institute of Standards and Technology (NIST) to assist members of the NIST staff, as well as others who may have need of such assistance, in the use of the SI in their work, including the reporting of results of measurements.

Aubrecht et al. (2011)
This is the second part in a series of notes that will help teachers understand what SI is and how to use it in a common-sense way. This part discusses units and the sorts of units that are part of SI, as well as the idea of coherence of SI

Ainslie (2015)
Introduction The current terminology of underwater sound, as documented, for example, by (Urick, 1983), was developed during and after the Second World War (ASA, 1951; Urick, 1967), and has evolved little since then (Jensen et al., 2011). When examined against a modern requirement, with particular attention to the needs of planetary oceanography and underwater noise, this 60-year old terminology is found wanting.

D’Amore (2015)
The present work is aimed at providing an engineering contribution to the study of tactile perception as regards surface texture, whose relevant information in terms of ‘coarseness’ is known to be carried in large part by friction-induced vibrations. Adopting an experimental approach focused on biomechanics, concurrent tribological and dynamical (vibratory) characterizations are carried out for a finger-surface sliding contact reproduced in vivo and in vitro; appropriate ‘contact indicators’ are provided, respectively in the form of ‘dynamic friction factor’ and in the form of both frequency-integrated ‘band vibration parameter’ and non-frequency-integrated ‘modulus of Fourier transform of friction-induced vibratory acceleration’ measured at the subject fingernail. Band vibration parameter is a custom frequency-integrated parameter defined within a selected frequency band and sensitive to the average power of friction-induced vibratory acceleration measured at the subject fingernail; specifically, it is calculated over four frequency bands related to the four populations of low-threshold cutaneous mechanoreceptors located within the volar glabrous skin of the human hand, as well as over the frequency band including all the previous, identified as ‘four-channel’ (FC) frequency band and ranging from 0.4 Hz to 500 Hz. The undertaken experimental campaign involves sixteen samples (six periodic metal samples, four isotropic metal samples, and six woven and knitted fabric samples), two subjects, and nine lubricants (one cream-like moisturizing product, two gel-like lubricating products, and six oil-like vegetable lubricating fluids). Operationally, a particular emphasis is put on the effect of moisturizer applied to the skin and on the effect of lubricant introduced between finger and surface, both in the absence and in the presence of an intermediate latex layer obtained from a natural rubber latex male condom. Dealing with a wide variety of experimental conditions and notably with dry, moist, and wet contacts, both direct and indirect, inter-subject variations are interpreted in terms of different water content of skin and prominence of papillary ridges, while inter-sample variations are interpreted in terms of different bulk properties (deformability) and surface properties (macroscale asperities). The effect of moisturizer on tribological response is variable among subjects, while on dynamical (vibratory) response it consists of a systematic attenuation of the average power of friction-induced vibratory acceleration, in agreement with psychophysical studies reporting a corresponding reduction of tactual subjective assessments of ‘coarseness’ perceptual attribute of ‘fine’ surface textures, as well as with neurophysiological studies reporting a corresponding reduction of neural activity within (especially fast-adapting) low-threshold cutaneous mechanoreceptors. The effect of lubricant on tribological response consists of a systematic reduction of dynamic friction factor, while on dynamical (vibratory) response it consists of a disappearance of the ‘threshold effect’ of sliding speed (i.e. a monotonic strong increase above a value of approximately 20 mms or 30 mms) and of a systematic and drastic attenuation of the average power of friction-induced vibratory acceleration, in agreement with psychophysical studies reporting a corresponding reduction of tactual subjective assessments of ‘coarseness’ perceptual attribute of surface textures. The attenuation and distortion of friction-induced vibratory acceleration observed in the presence of a lubricated or non-lubricated intermediate latex layer, while representing a first biomechanical confirmation of recent psychophysical results, have evident implications in terms of condom ‘acceptability’ since they appear to cause a significant and very early degradation of the mechanical signal which is believed to constitute a relevant component of the stimulation.

Slabbekoorn et al. (2019)
Offshore activities elevate ambient sound levels at sea, which may affect marine fauna. We reviewed the literature about impact of airgun acoustic exposure on fish in terms of damage, disturbance and detection and explored the nature of impact assessment at population level. We provided a conceptual framework for how to address this interdisciplinary challenge, and we listed potential tools for investigation. We focused on limitations in data currently available, and we stressed the potential benefits from cross-species comparisons. Well-replicated and controlled studies do not exist for hearing thresholds and dose–response curves for airgun acoustic exposure. We especially lack insight into behavioural changes for free-ranging fish to actual seismic surveys and on lasting effects of behavioural changes in terms of time and energy budgets, missed feeding or mating opportunities, decreased performance in predator-prey interactions, and chronic stress effects on growth, development and reproduction. We also lack insight into whether any of these effects could have population-level consequences. General “population consequences of acoustic disturbance” (PCAD) models have been developed for marine mammals, but there has been little progress so far in other taxa. The acoustic world of fishes is quite different from human perception and imagination as fish perceive particle motion and sound pressure. Progress is therefore also required in understanding the nature and extent to which fishes extract acoustic information from their environment. We addressed the challenges and opportunities for upscaling individual impact to the population, community and ecosystem level and provided a guide to critical gaps in our knowledge.

Where logarithmic units apply: neper and bel
I happened across the following (footnotes on Table 1-14 on page 1-35 of H. Wayne Beaty, SECTION 1 UNITS, SYMBOLS, CONSTANTS, DEFINITIONS, AND CONVERSION FACTORS): This suggests that the (deci)bel is not a formally correct unit for root-power (field) quantities, and conversely, the neper is not a formally correct unit for power quantities, ISO 80000-3 and general convenience notwithstanding. Since sources do not fully agree with each other, we should perhaps tone down the approach in this article that puts all cases on equal footing, along with a direct conversion between dB and Np. —Quondum 02:43, 22 January 2019 (UTC)
 * The decibel is defined for power ratios only. It may be applied to current or voltage ratios only when the resistances through which the currents flow or across which the voltages are applied are equal.
 * The neper is defined for current and voltage ratios only. It may be applied to power ratios only when the respective resistances are equal.
 * There's a difference between a formal definition (of a national or international standards body) and the preferred definition of individual scientists and engineers. I agree that some scientists, including several cited in the decibel article, believe that the decibel SHOULD be reserved for quantities proportional to power, but the simple fact is that it is used in multiple ways not compatible with this simple wish. I have an article somewhere that explains WHY the decibel should be reserved for power quantities. I will look it up and post the details. Dondervogel 2 (talk) 08:46, 22 January 2019 (UTC)
 * The article is by Mills and Morfey . Its abstract reads

"The use of special units for logarithmic ratio quantities is reviewed. The neper is used with a natural logarithm (logarithm to the base e) to express the logarithm of the amplitude ratio of two pure sinusoidal signals, particularly in the context of linear systems where it is desired to represent the gain or loss in amplitude of a single-frequency signal between the input and output. The bel, and its more commonly used submultiple, the decibel, are used with a decadic logarithm (logarithm to the base 10) to measure the ratio of two power-like quantities, such as a mean square signal or a mean square sound pressure in acoustics. Thus two distinctly different quantities are involved. In this review we define the quantities first, without reference to the units, as is standard practice in any system of quantities and units. We show that two different definitions of the quantity power level, or logarithmic power ratio, are possible. We show that this leads to two different interpretations for the meaning and numerical values of the units bel and decibel. We review the question of which of these alternative definitions is actually used, or is used by implication, by workers in the field. Finally, we discuss the relative advantages of the alternative definitions."
 * Dondervogel 2 (talk) 08:53, 22 January 2019 (UTC)


 * Perhaps we should start a new section in the article on RS opinions that differ from the standard definition(s). Dondervogel 2 (talk) 09:31, 22 January 2019 (UTC)
 * I'm less concerned with diversity of opinion than I am about misrepresenting what one might term the mainstream position. As an encyclopaedia, we should watch out for inadvertently painting a black-and-white picture of a grey landscape, in particular in a way that contributes to clouding the metrology.  It is evident that the practice with regard dB includes diverse uses, including the use in telephony of use of the unit dBm for a voltage as having as reference the voltage that would result in 1 mW of power being dissipated in a 600 Ω resistor if applied to it.  In camera sensor elements, the definition is even worse.  In short, the real-world practice is a definitional mess, and is only amenable to separate standardization within narrow disciplines.  ISO 80000 may have tried to extract a coherent picture, but failed, since not even the CGPM (is this not also the body that delegates the ISQ its authority?) accepts its definition: CGPM provides (via its and its daughter organizations' publications) a definition in the Draft Ninth SI brochure (and the 8th) that flatly contradicts ISO 80000, on both the bel and the neper, yet the article makes no mention of this contradiction.  My preference would be to present first the areas on which they all agree (bel as unit of power level, neper as a unit of amplitude level), and to move the murky/inconsistent bits to a section that presents the variations of definition from different sectors, both standards bodies and RS authors.  —Quondum 14:49, 22 January 2019 (UTC)
 * I agree it makes sense to provide a brief overview of the multiple ways the decibel is used in practice (referring reader to Decibel for detail. I don't think it's correct to say that the ISQ derives any authority from CGM. Rather the ISQ is an invention of ISO and IEC, working together, filling a gap left by CGPM in not defining physical quantities, and their corresponding units where these are outside the SI. CGPM is silent on the matter because the dB is not an SI unit. Dondervogel 2 (talk) 16:31, 22 January 2019 (UTC)
 * Okay, I'm functioning on pure guesswork, and getting it a bit wrong. But I'm not embarrassed at my confusion: The CGPM has authority over the CIPM, which in turn directs the BIPM, which publishes the VIM, which mentions (and apparently defines) the ISQ and the quantities that it defines.  If one sticks entirely to these (and ignores ISO 80000), a set of quantities and units with regard to level are defined, including the neper and bel, even though these are not classified as SI units (which is to say, I don't agree that the CGPM/CIPM/BIPM are silent on these non-SI units).  Given a bit of time, I'll try to work on the respective articles clarifying what comes from where, giving lower prominence to aspects that differ according to source.  I think that it is at least capturing that in some respects the picture is just confused.  —Quondum 20:23, 22 January 2019 (UTC)
 * We agree on your basic point, which is that the article would benefit by clarifying that multiple interpretations of the meaning of a level in decibels (or nepers) exist. By the way, do I understand you correctly that international standard definitions exist of dB and Np outside of ISO/IEC 80000? If so, where can I find these? Dondervogel 2 (talk) 21:21, 22 January 2019 (UTC)

Quoted from the current version: "(g) The statement L$A$ = n Np (where n is a number) is interpreted to mean that ln(A$2$/A$1$) = n. Thus when L$A$ = 1 Np, A$2$/A$1$ = e. The symbol A is used here to denote the amplitude of a sinusoidal signal, and L$A$ is then called the neperian logarithmic amplitude ratio, or the neperian amplitude level difference. (h) The statement L$X$ = m dB = (m/10) B (where m is a number) is interpreted to mean that lg(X/X$0$) = m/10. Thus when L$X$ = 1 B, X/X$0$ = 10, and when L$X$ = 1 dB, X/X$0$ = 10$1/10$. If X denotes a mean square signal or power-like quantity, L$X$ is called a power level referred to X$0$."

Quoted from the presumed future version: "Table 8 also includes the units of logarithmic ratio quantities, the neper, bel and decibel. They are used to convey information on the nature of the logarithmic ratio quantity concerned. The neper, Np, is used to express the values of quantities whose numerical values are based on the use of the neperian (or natural) logarithm, ln = log$e$. The bel and the decibel, B and dB, where 1 dB = (1/10) B, are used to express the values of logarithmic ratio quantities whose numerical values are based on the decadic logarithm, lg = log$10$. The statement L$X$ = m dB = (m/10) B (where m is a number) is interpreted to mean that m = 10 lg(X/X$0$). The units neper, bel and decibel have been accepted by the CIPM for use with the International System, but are not SI units."

In the 8th edition, examples are given using amplitude and power. In the draft 9th edition (which I presume supersedes the 8th on 20 May 2019), they drop these examples, making no reference to specific type of quantity. The VIM does refer to ISO/IEC 80000, but without directly deferring to it. —Quondum 00:25, 23 January 2019 (UTC)
 * Hmmm ... these are examples of how logarithmic units are to be used, and I agree that is relevant, although in the 9th edition even that is gone. What I do not see is a definition of either neper or bel. The existing ISO/IEC 80000 series is likely to be superseded in 2019, and when that happens I expect the definitions to be removed from ISO 80000-3 too. Perhaps they will re-appear somewhere else. If not, all we have left to guide us (apart from the countless incompatible ways they are used in the real world) is the limited advice of the SI brochure. We shall see. Dondervogel 2 (talk) 07:57, 23 January 2019 (UTC)
 * Depending on interpretation (specifically, on where the description applies), the SI text constrains the interpretation, and hence effectively defines the units. It does not claim to be the standard on it. Interesting that you expect the definitions to be removed from ISO 800000 – perhaps a symptom of the inability to standardize a mess?  Nevertheless, we can try to capture the use of these scales and units, and not treat it as a standards-driven concept.  —Quondum 14:57, 23 January 2019 (UTC)
 * I support your proposal but do not have time to work on it myself. I'm happy to comment on any specific improvements. Dondervogel 2 (talk) 08:49, 24 January 2019 (UTC)

The level of a quantity, also called a logarithmic ratio quantity, is the logarithm of the ratio of the value of that quantity to a reference value of the same type of quantity. A level may be considered to be a dimensionless quantity, but units are generally used and indicate the selected base of the logarithm and may depend on the nature of the quantities. The International System of Quantities and other standards bodies attempt to standardize usage. Examples of levels are the various types of sound level: sound power level (literally, the level of the sound power, abbreviated SWL), sound exposure level (SEL), sound pressure level (SPL) and particle velocity level (SVL).

Definitions
Definitions of a level or logarithmic ratio quantity and its units vary according to the standardizing body and the type of quantity.

Generally applicable definitions
The level of a power quantity Q with respect to a reference quantity Q$0$ may defined in terms of the unit bel (symbol B) or submultiples thereof as:
 * $$L_{Q/Q_0}=\log_{10} \frac{Q}{Q_0}\,\text{B}=10 \log_{10} \frac{Q}{Q_0}\,\text{dB}.$$

The level of a root-power quantity Q with respect to a reference quantity Q$0$ may defined in terms of the unit neper (symbol Np) as:
 * $$L_{Q/Q_0}=\log_\mathrm{e} \frac{Q}{Q_0}\,\text{Np}.$$

The restrictions above for each unit to specific types of quantity avoids ambiguities and inconsistencies, as may be seen from the sections below.

SI
The International Bureau of Weights and Measures (BIPM) in its SI Brochure provides the above definitions without restrictions, though the examples that it provides conform to the restrictions.

In its draft of the 9th edition of the SI Brochure, it provides equivalent definitions, but without specific examples and still without restrictions.

ISO 80000
The International Organization for Standardization standard ISO 80000-3 defines level and its units, but in a way that implies first converting a ratio to a corresponding ratio of power quantities (assuming a linear medium) when the unit is the bel (or any direct multiple thereof such as the decibel) is used.

Thus, when we say that L$e$ = 1 B, we have that
 * X/X$10$ = 10, if X is a power quantity
 * X/X$A$ = 100, if X is a root-power quantity

Accordingly, the category (power or root-power) of the quantity X must be specified for this definition to make sense. While in most disciplines (e.g. electronic engineering, sound engineering, RF engineering), the category of each quantity is uniformly understood, there are incompatibilities between disciplines.

ANSI
The American National Standards Institute defines a level with a rule for determining the base of the logarithm, which

The level of a quantity Q, denoted LQ, is defined by
 * $$L_Q = \log_r\!\left(\frac{Q}{Q_0}\right)\!,$$

where
 * r is the base of the logarithm;
 * Q is the quantity;
 * Q0 is the reference value of Q.

Discrepancies
The SI text implies that 1 dB = (ln 10)/10 Np ≈ 0.23026 Np, whereas ISO 80000-3 provides the conversion 1 dB = (ln 10)/20 Np ≈ 0.11513 Np. However, it is rare for the neper to be used for power quantities, and for the bel, if root-power quantities are treated as being referred to power levels, the latter equality would fall away, to be replaced by a correspondence.

Discussion of working attempt content
Please edit and comment on the "working attempt" freely. We could also move this section to a subpage if preferred. I know that it has an OR feel, but I think that at least the structure might be a start. —Quondum 23:41, 27 January 2019 (UTC)


 * Quondum, we usually work incrementally on the existing version, not on an out-of-sequence replacement draft. This helps editors see what direction things are changing, keeps an attribution history, etc.  I don't see how I can engage in the process you propose. Dicklyon (talk) 00:14, 28 January 2019 (UTC)


 * No problem; I will be happy to edit in my changes to the existing version. My hesitancy was because I do not know whether my approach above might be objectionable; if you do not find my attempts at clarifying the problems as per my observations earlier too jarring, I can proceed to do so.  —Quondum 00:30, 28 January 2019 (UTC)
 * Yes, you should probably proceed, but proceed incrementally, giving people time to react before getting in too deep. Dicklyon (talk) 05:13, 28 January 2019 (UTC)
 * In your proposed opening sentence, "also called a logarithmic ratio quantity" is misplaced; it would need to immediately follow what it refers to. That's as far as I got. Dicklyon (talk) 05:15, 28 January 2019 (UTC)
 * And instead of "or submultiples thereof", mention the decibel, the only submultiple ever used. Dicklyon (talk) 05:17, 28 January 2019 (UTC)
 * Thank you for doing this, Quondum. My comments follow, approximately in the order they arise
 * I suggest replacing Carey (2006) with Morfey (2000) . Morfey is more general (and in my opinion more scholarly and more mainstream) than Carey.
 * The ISO 80000 definitions are L_P = (1/2) ln(P/P_0) and L_F = ln(F/F_0), respectively.
 * I see no discrepancy between SI "definitions" of dB and Np and the definitions of ISO 80000.
 * Use of the term "amplitude quantity" is problematic, because (as far as I know) none of the standards mentioned use this term.
 * I applaud your objective (to explain how levels and their units are used in practice) but do no see how these changes help meet that objective. The Mills and Morfey article might be helpful in this regard.
 * Sorry I can't be more pro-active but I'm completely swamped. Dondervogel 2 (talk) 08:30, 28 January 2019 (UTC)


 * I appreciate the feedback. I may need time to source the Morfey text. I would like to see eye-to-eye on one thing before proceeding. Dondervogel's comment "I see no discrepancy between SI 'definitions' of dB and Np and the definitions of ISO 80000" is close to the heart my concern.  Two statements from the standards that I see as incompatible are:
 * SI: The statement L$2$ = m dB = (m/10) B (where m is a number) is interpreted to mean that m = 10 lg(X/X$1$).
 * ISO: L$A$ = 10 lg ($2$)$1$ dB
 * Putting X = F produces incompatible formulae, which could be called a discrepancy. I was hoping to at least delineate where the terminology is on firm ground, as per the quote from Wayne Beaty with which I started this thread, and where caution in interpretation is needed, standards notwithstanding.  —Quondum 01:45, 29 January 2019 (UTC)
 * Thank you for explaining. I agree those two statements are incompatible, but it does not follow from the SI text that 1 dB = 0.23026 Np. Instead I would argue that the Np and dB are independently defined, with no special relationship between them, with Np applying to amplitudes (distinct from root-powers) and dB applying to powers. This is precisely the position of Mills and Morfey (2005), suggesting that Professor Mills had a stronger influence on BIPM than on ISO. Dondervogel 2 (talk) 09:48, 29 January 2019 (UTC)
 * I would agree with your assessment of the SI statement if they had indicated a restriction on the type of quantities in the ratio, but as it is, it is a statement about general ratios. Of course, since it does not purport to be a standard on the matter, we can choose to treat this as an oversight.
 * If qualified by a restriction to quantities of specific types of ratios, namely power for dB and amplitude for Np (and I like the distinction between amplitudes in a strict sense (i.e. referring to a multiplier of a waveform, not to an averaged quantity), the SI statement is equivalent to what you have just said. Without the restriction (i.e. taken to apply to every ratio), I contend that it does imply that 1 dB = 0.23026 Np.  However, I would argue that the restriction is necessary to make the SI statement compatible with actual usage, so please regard it only as an illustration of the poor quality of the wording in the standards.  ISO 80000, in contrast, makes an explicit equivalence (1 dB = 0.11513 Np) that just breaks everything, and thus to use it as the core of a description will only entrench a bad position, that I suspect we would find is a minority position, despite being a standard.  Restriction of usage to types of ratios is standard in everyday usage: e.g. we never refer to the log ratio of frequencies in dB, or even to the log ratio of angles (which is the ratio of the integral of the frequencies over time) as octaves.  Extending this, the only real-world violations of such restrictions on units of level that I am aware of is the dB (and the Np if we include ISO 80000!), and the dB can be reconciled by careful wording, namely by referring the quantity ratio to a power ratio in a linear medium: we would not formally refer to voltage gain, but to the power gain in dB of the voltage in a reference medium.  It is the careful restriction to ratios of specific types of quantity that we are finding support for in some sources, and it is this that I feel the article could benefit from, as well as being less standards-centric.  —Quondum 13:54, 29 January 2019 (UTC)
 * I don't have time to read carefully (am travelling) but it sounds like we agree on most aspects of this. Reading your comment about discrepancies reminds me of an IEEE standard that has its own version of these equations. I'll look up the details. Dondervogel 2 (talk) 18:02, 29 January 2019 (UTC)
 * See p 26-27 of IEEE SI 10-2016 'American National Standard for Metric Practice'. Dondervogel 2 (talk) 18:15, 29 January 2019 (UTC)

Q, I haven't read all your points in detail, but I'm worried where you say "ISO 80000 ... makes an explicit equivalence (1 dB = 0.11513 Np) that just breaks everything". How does this break anything? It seems right to me, while "1 dB = 0.23026 Np" can't possibly work. Yes, the restrictions of formulae to the right types of quantities is implicit in making all this work, and is made explicit in some sources. So maybe we can say where it is explicit and where it is implicit. In practice, there's little barrier to moving bertween dB and Np since their usage is almost always in the context of systems that are presumed linear; or at least the "loads" are presumed linear. Dicklyon (talk) 18:24, 29 January 2019 (UTC)
 * Dondervogel, don't feel pressured – I'll be slow anyway; besides, I still need to visit my local library to see what I can find.
 * Dicklyon, oxhides and decibels have similar problems as units of measurement. Rather than trying to argue the case, let me see what additional sources I can find to paint a clearer picture.  I opened this thread in surprise when I realized that what seems obvious to me might have solid support in the literature.  —Quondum 02:29, 30 January 2019 (UTC)
 * I managed to lay my hands on a copy of IEEE/ASTM SI 10-2016. Its approach to resolving the issue is subtly different: it defines two different quantities – a "field level" or "level-of-field" of a quantity and a "power level" or "level-of-power" of a quantity.  In my mind, this is like the difference between "diameter" and "radius" – inherently distinct quantities.  Rather unfortunately, the same symbol is used: a subscripted italic L.  Here, I'll use nonitalic F and P respectively: F$A$ and P$X$.  The units Np and dB are assigned for both (again somewhat unfortunately, and I'll differentiate by using primes here).  This all makes sense in a way, as long as we are explicit about which type of level we are referring to (i.e. we do not infer it from the type of the quantity). An interesting observation: the neper can be argued to be the coherent unit of amplitude gain, but in its use for power levels (or for frequency levels), the same argument does not apply.
 * It would be fair to say using this definition,
 * The power level of X with reference X$0$ is P$X$ = 10 lg(X/X$0$) dB = $X$ ln(X/X$0$) Np″.
 * The field level of X with reference X$1/10$ is F$X$ = 20 lg(X/X$0$) dB′ = ln(X/X$X$) Np.
 * One can then say that 1 dB = 0.11513 Np″ and 1 dB′ = 0.11513 Np. It is tempting to equate dB = dB′ and Np = Np″, giving P$0$ = F$0$ (or vice versa), but this makes no more sense than saying 1 oct = ln(2) Np and we lose the more beautiful equation P$X$ = F$0$ + F$F$ (the power level-of-power is the sum of the voltage level-of-field plus the current level-of-field).
 * Coming back to earth (and putting my analytic perfectionism aside), IEEE's approach of not defining a level generically and rather defining a field level and a power level separately makes sense and closely matches practice. Using the same symbol (L$F⁄F0$) and the same units (dB and Np) for both types of level remains unfortunate IMO.  I'll think about how to adjust the article so that it does not jar for any of us.  —Quondum 00:39, 3 February 2019 (UTC)
 * Aside: This allows us to disambiguate between the image intensity "level-of-field" and image intensity "level-of-power" ;)  —Quondum 01:01, 3 February 2019 (UTC)
 * There are lots of unfortunate things going on, I agree. That image intensity one can't be sensibly reconciled, I think. Dicklyon (talk) 02:39, 3 February 2019 (UTC)

I obtained. In a nutshell, it makes a powerful argument that for units, the choice of coherent unit is arbitrary (including, specifically, for logarithmic and angle quantities): it all depends on the quantity's defining equation. It then argues that it provide defining equations so that Np and rad are the coherent units is mathematically convenient. My take: (a) the latter is equivalent to arguing that the coherent unit of length should be, and (b) the former I take as a is a good argument why we should not say Np = 1 or rad = 1, and why angle and logarithmic decay must be base quantities, which is a corollary that the authors seem to have missed. As far as this article is concerned, my takeaway is: field levels and power levels are quantities of different types, and the concept of "level" as a quantity is not sensible (in the sense of "Power level and field level are examples of the quantity 'level'"). —Quondum 19:23, 18 February 2019 (UTC)
 * I agree that power level and field level are quantities with quite different properties, and for this reason it would be helpful if they had different units (eg dB and Np, respectively). I also think that defining the decibel in terms of the neper is a really dumb idea, for the same reason. What I don't see is why they can't both be seen as examples of the more general concept of level, defined as L_Q = log_r(Q/Q_0). As previously stated, this is the ANSI definition of level, and includes quantities as diverse as frequency level (in octaves), grain size (in phi units) and pH.
 * The snag is that the world is not how I would like it to be. The decibel is defined in terms of the neper, and level of a field quantity is defined as the level of the square root of a power quantity, making it IMHO completely redundant.
 * I'm not sure where this leaves us though. What specific change would you like to see? Dondervogel 2 (talk) 22:18, 18 February 2019 (UTC)
 * Snap w.r.t. the "snag". Using VIM terminology ("quantities of the same dimension are not necessarily of the same kind"), we are saying that 'level' is at best a category of quantities, not all of the same kind.  What I would like to see: agreement on a clear definition of the topic of this article (that I understand).  Since you do not seem convinced that 'level' and 'logarithmic ratio quantity' are the same thing, what is the difference?  Is it that the denominator of the ratio must be a standard reference value (thus making a relative level not a level)?  Or something else?  What is the frequency level of middle C?  —Quondum 23:30, 18 February 2019 (UTC)
 * According to ANSI S1.1-2013 the frequency level of C_4 is 48 semitones (i.e., 4 octaves). The reference value used for the musical scale is C_0 (approximately 16.4 Hz) Dondervogel 2 (talk) 23:58, 18 February 2019 (UTC)
 * In the present context I think level and logarithmic quantity are exactly the same thing. I just don't expect others to agree. This is why I introduced the "litmus" test: If we can agree that pH is a level, then it can be included in this article, and then I agree with changing the name in the way you have suggested; but if pH is not a level, then it cannot be included in this article, and in that situation the name should be left as it is now. Am I making sense? Dondervogel 2 (talk) 00:06, 19 February 2019 (UTC)
 * I wonder whether that was invented by the ANSI standard. "A frequency level of 4 octaves" does not strike me as something someone would understand, as opposed to "a frequency level 4 octaves above C$2$". Which is to say, I understand the octave as a unit of relative level (interval), not of absolute level (pitch).
 * I would prefer it if you did not use the word "level" in trying to clarify the scope of the article. It only makes sense to speak of how it is used (i.e. with what meaning it is used by whom), not of what it "is" or whether something "is a level" without giving the specific context, because its use evidently varies substantially by context.  —Quondum 00:43, 19 February 2019 (UTC)
 * Frequency level was not invented by ANSI. In fact it was used before ANSI even existed. ANSI just formalised the practice of the time, although I suspect it has now fallen into disuse.
 * I don't see what's stopping us creating an article Logarithmic ratio alongside this one. With the new article in place we can then discuss whether to merge the two.
 * Dondervogel 2 (talk) 11:38, 19 February 2019 (UTC)
 * I'm starting to think that would be indistinguishable from Logarithmic scale (well, it would be broader, since scales are "absolute"). Since I am unable to get a clear idea of any preference from you for a scope boundary for this article, let me propose something: logarithmic scales to include in this article are those related to power, field and amplitude quantities: exactly those for which the units dB and Np are used.  I find it challenging to define a category boundary between this and [quantities measured on] logarithmic scales. —Quondum 15:28, 19 February 2019 (UTC)
 * I think the distinction is that quantities expressed in decibels and nepers are usually referred to as "levels", whereas those expressed in other units (including frequency level) are usually not. This is why I'm reluctant to change the title from the present one. Dondervogel 2 (talk) 16:30, 19 February 2019 (UTC)
 * Then let's agree to restrict the scope to these quantities. Then, at least, I will be able to think about the content coherently.  Frequency level would be removed from the article, and the use of the ANSI definition within the article would be restricted to the types of level that the article covers.  —Quondum 18:04, 19 February 2019 (UTC)
 * Not so fast. The fact remains that frequency level *used* to be referred to as a level, but I accept no longer (except in that one ANSI standard, a remnant of times gone by). That means it can be removed from the main body but still belongs in a historical introduction to the article. Dondervogel 2 (talk) 19:27, 19 February 2019 (UTC)
 * An article is about a concept, not a term. Context (history, related concepts, similar names, etc.) can be given, without being confined to the scope, which should satisfy what you say.  The name of the article does not determine the scope; it is the other way around.  Anyhow, this restriction will make it easier to work on the article.  —Quondum 19:49, 19 February 2019 (UTC)
 * I like that the idea of concept first. I'd not thought of it that way before. I'm happy for you to lead and I'll follow as best I can. The only other editor who has shown any interest is Dicklyon, and we should check he's on board. Dick? Dondervogel 2 (talk) 21:45, 19 February 2019 (UTC)
 * Sounds reasonable. Also, you said "frequency level *used* to be referred to as a level"; where/when was that? Dicklyon (talk) 05:40, 20 February 2019 (UTC)
 * The term was introduced by Fletcher in 1934. It was used in the 1930s, 40s and 50s. The term was later standardized by ANSI but seems to have fallen into disuse . Dondervogel 2 (talk) 07:55, 20 February 2019 (UTC)
 * Actually, it's not hard to find recent examples in musical or speech acoustics. Dondervogel 2 (talk) 10:40, 20 February 2019 (UTC)
 * Thanks for those! I had not realized.  So do you want to keep frequency level in, even though frequency is not related to a power or root-power quantity? Dicklyon (talk) 16:02, 20 February 2019 (UTC)

Discussion of working attempt content (cont)
Well, that was my initial position, but Quondum makes a strong case for reconsidering. He points out (correctly) that my thinking was back to front. In a nutshell, my reasoning was "the article is called 'Level' so it encompasses all terms of the form 'this level' or 'that level'". Instead it should be "we choose a scope X and because of that scope 'this level' is included and 'that level' is not". If the scope is limited to levels traditionally expressed in dB or Np (which is my understanding of Quondum's proposal - he can correct me if I've got this wrong), that would exclude frequency level. Dondervogel 2 (talk) 17:03, 20 February 2019 (UTC)
 * Yup, I'm suggesting a scope that includes power-related levels: field level and power level. Since the root-power/field levels are so closely intertwined with power levels that they traditionally share a unit, sharing an article should work.  The rest (other logarithmic quantities) are just "related" – sharing a larger category.  A section could deal with the special use of Np for decay and propagation, where it ties in with rad.  —Quondum 18:23, 20 February 2019 (UTC)
 * I like that, especially the connection between Np and rad. Dondervogel 2 (talk) 19:36, 20 February 2019 (UTC)
 * OK, I can understand that motivation to go that way. But what could we call it, and would this this be an idiosyncratic way to split the meaning of "level"?  If instead we generalize a bit to logarithmic ratio quantities, we could discuss levels of various sorts within that, separating out power levels into a section, perhaps.  I'm open. Dicklyon (talk) 04:17, 21 February 2019 (UTC)
 * There are what I think of as three meanings that should be covered: power levels, root-power levels and finally the closely related gain/attenuation quantities. As a title, perhaps "Power and root-power levels"?  These are three or so distinct quantity types, with different defining equations.  Gathering the first two together into an article makes sense, and the third would maybe be a stub section referencing Propagation constant.  The first two are really the scope (and which usually are measured in dB), whereas the last clarifies the relationship with the Np, how they are considered to be related, and how the NP in turn relates to the rad.  Or that is what I am thinking at the moment, anyhow.  —Quondum 13:54, 21 February 2019 (UTC)
 * It sounds like your proposed scope could be summarized as "levels and level differences usually expressed in decibels". That would also include gain (e.g., amplifier gain) and attenuations (e.g., transmission loss). Dondervogel 2 (talk) 17:14, 21 February 2019 (UTC)
 * Close. Or perhaps we should make it exactly that, and relegate things that are usually primarily measured in nepers to being referenced in a section on related concepts and consider them outside the primary scope.  This kind of makes sense: though intrinsically logarithmic quantities, quantities with units Np and rad when used in propagation and decay (I think) do not get called levels, or even a difference in level (as is gain).  When talking of a decay constant of 0.5 Np/s, we are not interested in the change in level from 1.6 Np to 0.6 Np in 2 s, but rather in the slope (0.5 Np/s) itself.  This is even more the case for propagation constant, e.g. -0.5 Np/s + j 57.9 rad/s.  —Quondum 17:37, 21 February 2019 (UTC)

Rename to "Logarithmic ratio quantity"?
I propose a renaming: it seems to me that the better name for this article is Logarithmic ratio quantity, currently a redirect to here. Logarithmic quantity is an alternative, currently a redirect to Logarithmic scale. Feelings? —Quondum 23:37, 8 February 2019 (UTC)
 * It is an unambiguous name for all quantities of this type, even though their names might be various, for example frequency level and [Interval (music)|interval]], and probably is more immediately meaningful to most readers not very familiar with the topic.
 * Unlike Level (logarithmic quantity), it does not need disambiguation in the name.
 * The term level competes for unrelated everyday uses, e.g. sea level, and innumerable "level of ..." phrases.
 * The standards seem to use this term as a general heading:
 * SI uses the term logarithmic ratio quantity exclusively, and never uses the term "level" in this sense.
 * ISO 80000-3 refers to "logarithmic quantities" when it is being general, including in a heading (e.g. "logarithmic quantities and their units").
 * IEEE SI 10-2016 uses the heading "Logarithmic ratio quantities" under which to discuss these quantities, referring to field level and power level as instances.
 * Though ISO 80000 does use level as a semi-standalone term (but essentially as a shorthand for level of ... when it is clear which quantity is being used), IEEE SI 10 does not, always using the full terms field level and power level.
 * Good question. My instinctive reaction is to ask whether pH is a level. If the answer is "yes" we should add pH to the article and then I agree with your proposed change. If the answer is "no" I think the title is best left unchanged. Dondervogel 2 (talk) 00:22, 9 February 2019 (UTC)
 * I guess you could call that a litmus test. Dondervogel 2 (talk) 00:25, 9 February 2019 (UTC)
 * *groan* The logic in your response is difficult to parse.  It is easier to ask whether pH is a logarithmic quantity, and I would contend that it is (albeit with a base that is less than 1: "In chemistry, a decimal cologarithm is indicated by the letter p.").  And no-one would object to the term "pH level", though the standards we have listed seem to be silent.  Perhaps my question is whether the article should be allowed to expand its scope slightly, and I contend that although the standards use "level", they (except possibly ANSI/ASA S1.1, which I do not have) do not define it in a sufficiently stand-alone way to merit an article, whereas the other terms (like logarithmic ratio quantity) are defined well.  But we will have to consider overlap with Logarithmic scale.  —Quondum 02:09, 9 February 2019 (UTC)
 * Sorry, I didn't mean to be cryptic. I agree with you that pH is a logarithmic ratio quantity. I don't consider this to be contentious, but is it also a level?
 * I think the answer to both this question and yours hinges on whether we accept the ANSI definition of "level". If we do then all logarithmic ratio quantities (including pH) are also levels, and vice-versa. In other words "level" and "logarithmic ratio quantity" are synonyms.
 * I used pH as an example because I think it provides a good test case: If we consider pH to be a level, this implies that we accept the ANSI definition of level and the above logic follows, and not otherwise.
 * Does this help? Dondervogel 2 (talk) 20:17, 9 February 2019 (UTC)
 * I think this is helpful in getting clarity. We have, in effect, a few questions:
 * Do we consider "level" and "logarithmic ration quantity" to be synonymous?
 * Should we expand the scope of the article to cover "logarithmic ratio quantity"?
 * Should we rename the article as suggested?
 * My thinking goes as follows: we cannot answer the first question with confidence (we don't know for sure whether level as a standalone concept as per ANSI is notable or even accepted generally), so it is wise to change the scope to something closely related that is well-defined and notable. That would suggest that we expand and rename the article, include pH and similar, and use the term "level" with only those quantities in the article where this is the standard terminology.  This synergizes with SI and IEEE, ISO fits either way, and allows us to mention that ANSI defines the term level in the more general sense.  —Quondum 01:20, 10 February 2019 (UTC)
 * I see. I quite like the present scope myself, which is that of the title (level). I think the concept of level causes confusion and having such an article helps counter that confusion. Broadening the scope to logarithmic ratios generally would mean losing this focus and lead to a large overlap with Logarithmic scale. One option might be to create a new article Logarithmic ratio quantity alongside this one, and we could decide whether to merge them later. What do others think? Dondervogel 2 (talk) 10:12, 10 February 2019 (UTC)

Don't we need two separate articles?
It seems to me that if we follow Quondum's proposal (and I think we should) we're going to need two separate articles: One covering all logarithmic quantities (including frequency level and pH); the other limited to logarithmic quantities usually expressed in decibels. Dondervogel 2 (talk) 15:19, 23 February 2019 (UTC)


 * I agree that these are two topics and should be in separate articles. What is less clear is whether an article covering logarithmic quantities should be separate from Logarithmic scale, which I don't feel compelled to settle at this point, as long as the scope of this article is clear.  —Quondum 18:26, 23 February 2019 (UTC)
 * The way Logarithmic scale is written now it is more about the concept of visualizing linear quantities on a logarithmic scale than about logarithmic quantities per se. I think there's room for both. My thinking is driven by the question of where to put the material on frequency level if removed from here. Dondervogel 2 (talk) 19:46, 23 February 2019 (UTC)
 * I tend to agree. And yes, before we delete it here, we need a home to move it to.  So I think it can remain here in a (temporary?) "related quantities" section retained for the purpose until we sort that out.  —Quondum 20:48, 23 February 2019 (UTC)
 * Yes, I think it would need to be temporary. It doesn't really fit in your proposed scope. And sorry for being a bit slow. When I think back I can see you've been arguing for this all along. I think I've finally caught up with your thinking. Dondervogel 2 (talk) 22:01, 23 February 2019 (UTC)
 * Same here. I'm OK with the proposal, and don't much care where frequency level ends up. Dicklyon (talk) 22:24, 23 February 2019 (UTC)

Quantities and their proxies
This may be is a bit of a philosophical ramble, so read it only if you are feeling philosophical! The argument on the arbitrariness of quantities by Mills et al (2001) is what prompts this (I unfortunately don't have Mills & Morphy (2005)).

In metrology, we look for quantities that sometimes act like vectors to work with: when we add them, under some conditions this mirrors what happens to the real world. Lengths, angles, gains all fit this characterization. Where any scale factor would do, we use units to give us a scale. That is the basis of metrology. Natural units try to find a "best" unit for each, such that universal constants take a particularly simple form when expressed in theses units, but this still does not remove the units (it is a mistake to say c = 1: it is really c = 1 lP/tP, or whatever). In this sense, angle is really not dimensionless: it has to be defined relative to some unit. If I say the angle is 10°, there is no ambiguity; the unit degree is well-defined. Even though I cannot know what unit of angle really is equal to a dimensionless 1 (because it depends on my choice of proxy), an angle of 10° remains unambiguous as an angle, no matter what scale factor I use to define the proxy of angle, because the unit takes care of that.

The problem I'm heading for is the definition of the quantity "power level". If we define unitless proxies of angle and of level as in SI and ISO, we are saying that the the different proxies are different quantities. And then there is no contradiction in saying that bel = 1 as a unit of the quantity log10P/P0 and neper = 1 as a unit of the quantity $X/X0$logeP/P0, because they are different quantities, and of course, since the units are just 1, algebraically neper = bel. How ISO resolves this is to say that it is incorrect to call log10P/P0 a power level. SI is kind of confused on this anyway.

Now for another approach: treat both the above dimensionless quantities as only proxies (each being related by a fixed factor) of a quantity we call power level. When P/P0 = 10, LP = 1 bel. When P/P0 = e, LP = 0.5 Np. This makes a conversion of units well-defined, fits with what we mean by "quantity", and does not force us to choose a base for the logarithm. The key here is to distinguish the quantity power level from a unitless proxy. A power level is unambiguously defined, the units are unambiguously defined, our intuition is not led astray by confusion, and the world is a happy place, as long as we accept that we have a true new base quantity (and associated base unit).

You will note that the article is written with the above intuition in mind, and I have just removed the equating of the units to unitless quantities, and it can be seen to be natural and understandable. I intend to re-insert that the standards equate the units to unitless numbers, but as a separate section. —Quondum 15:56, 24 February 2019 (UTC)


 * That all sounds great, and I agree that the way level is handled in the standards is confusing, misleading, or worse. But what do you have for sources that do it better, along the lines you suggest? What is this Mills and Morphy thing you refer to? Dicklyon (talk) 20:10, 24 February 2019 (UTC)
 * Quondum refers to a paper by those authors entitled On logarithmic ratios and their units. The abstract reads

"The use of special units for logarithmic ratio quantities is reviewed. The neper is used with a natural logarithm (logarithm to the base e) to express the logarithm of the amplitude ratio of two pure sinusoidal signals, particularly in the context of linear systems where it is desired to represent the gain or loss in amplitude of a single-frequency signal between the input and output. The bel, and its more commonly used submultiple, the decibel, are used with a decadic logarithm (logarithm to the base 10) to measure the ratio of two power-like quantities, such as a mean square signal or a mean square sound pressure in acoustics. Thus two distinctly different quantities are involved. In this review we define the quantities first, without reference to the units, as is standard practice in any system of quantities and units. We show that two different definitions of the quantity power level, or logarithmic power ratio, are possible. We show that this leads to two different interpretations for the meaning and numerical values of the units bel and decibel. We review the question of which of these alternative definitions is actually used, or is used by implication, by workers in the field. Finally, we discuss the relative advantages of the alternative definitions."
 * The paper argues that the decibel and the neper are units of two conceptually different quantities, and that no benefit arises from linking them. They make a very valid point but the paper is not widely cited.
 * Dondervogel 2 (talk) 20:23, 24 February 2019 (UTC)


 * Thanks. With the correct author spelling Morfey it's not as hard to find.  I'll email an author to see if a copy is available.  Dicklyon (talk) 21:07, 24 February 2019 (UTC)
 * I guess I should try to get this reference (I have a friend with access to Metrologia, and have been leaning on this friendship). Unlinking the different types of level and giving them different units makes sense, but I'd like to see what they have to say about it.  This does not mean that they will support my thesis, but they may make a similar point in order to unlink them.  There is a way around the question, even without sources: presenting only the cohesive bits that always make sense and tend to be common in the main section, and leaving the bits that confuse for separate mention, as I have done.
 * It is interesting to note that ISO 80000-13:2008 presents similar quantities and units without suggesting that they are dimensionless, in the style currently in this article:
 * {| class="wikitable"


 * 13-24 || information content || I(x) || I(x) = lb $X/X0$ Sh = lg $0$ Hart = ln $X/X0$ nat where p(x) is the probability of event x || See ISO/IEC 2382-16, item 16.03.02. See IEC 60027-3.
 * 13-24.a || shannon || Sh || value of the quantity when the argument is equal to 2 || 1 Sh ≈ 0,693 nat ≈ 0,301 Hart
 * 13-24.b || hartley || Hart || value of the quantity when the argument is equal to 10 || 1 Hart ≈ 3,322 Sh ≈ 2,303 Hart
 * 13-24.c || natural unit of information || nat || value of the quantity when the argument is equal to e || 1 nat ≈ 1,433 Sh ≈ 0,434 Hart
 * }
 * —Quondum 21:25, 24 February 2019 (UTC)
 * 13-24.c || natural unit of information || nat || value of the quantity when the argument is equal to e || 1 nat ≈ 1,433 Sh ≈ 0,434 Hart
 * }
 * —Quondum 21:25, 24 February 2019 (UTC)

A nice paper on this topic by Ian Mills (2002): On logarithmic ratio quantities. —Quondum 20:43, 18 May 2020 (UTC) PS: Nutshell: The quantities Logarithmic amplitude ratio (unit neper) and Mean square signal level and power level (unit bel or decibel) are not the same type of quantity. —Quondum 21:00, 18 May 2020 (UTC)

ISO 80000-3:2019?
Has anyone noticed that the status of the 2019 revision of ISO 80000-3 has changed from 'Approval' to 'Publication'? I believe this means that ISO 80000-3:2006 will be withdrawn soon, probably by the end of April. This and closely related articles (e.g., Decibel) will need modification to take into account the associated changes. Dondervogel 2 (talk) 22:14, 7 April 2019 (UTC)


 * Nope, not I. I'll have to try to get my hands on a copy.  It might make sense, in the interim, to edit ISO 80000-3 to mention the publication with this URL.  —Quondum 00:11, 8 April 2019 (UTC)
 * Better to wait until it reaches "published" – it seems still to be "under publication". —Quondum 00:26, 8 April 2019 (UTC)
 * ISO 80000-3:2006 has now been withdrawn, and replaced with ISO 80000-3:2019. The definitions of "level" and "decibel" are omitted from the revised standard, which has implications for this article. Dondervogel 2 (talk) 21:50, 23 October 2019 (UTC)
 * Wow. They've really trimmed it down – perhaps they realized that they had exceeded their mandate on earlier revisions, and indeed IMO were only muddying the waters by trying to capture the inconsistent practices of several industries.  Yes, I guess this will have implications for this article and others, though I'm not sure exactly what.  There are still numerous sources that define quantities called "level".  The 9th SI brochure still defines a simpler concept "logarithmic ratio quantity" and the units dB and B, and mentions the unit Np without clearly defining it.  Interesting.  —Quondum 01:04, 24 October 2019 (UTC)
 * Sounds good to me – "logarithmic ratio quantity" is at least pretty definite, whereas "level" is used informally for all sorts of things, logarithmic or not. But why omit decibel? Dicklyon (talk) 03:05, 24 October 2019 (UTC)
 * The plan, as I understand it, is to develop a new standard (IEC 80000-15) defining units of logarithmic quantities (dB, Np, oct, dec, pH, etc), all in one place. I assume that the definitions of dB and Np were withdrawn to give the developers of part 15 a free hand. They have their work cut out and I wish them luck. Dondervogel 2 (talk) 13:14, 26 October 2019 (UTC)
 * Surely not with the name "IEC 80000-15: Telebiometrics related to telehealth and world-wide telemedicines"? Or was that a plan that has been replaced by something else?  I agree that units of logarithmic quantities will be a challenge to standardize without inconsistencies.  —Quondum 17:15, 26 October 2019 (UTC)
 * Their provisional name seems to be Quantities and units – Part 15: Logarithmic and related quantities, and their units. They don't seem to have made much progress so far. I hope they start simple. If they can just agree on a definition for (say) octave and decade that would provide a valuable framework for other units. Dondervogel 2 (talk) 18:38, 26 October 2019 (UTC)
 * Nice. I'll be interested to see what they produce.  Before they get to any units (even the simpler cases octave and decade), they will have to give a clear answer to the question "In what way is a logarithmic unit restricted with respect to the kind of quantity in the ratios to which they apply?".  I also hope they end up effectively making logarithmic units into quantities of a different kinds (thus dropping identities like Np = 1).  Subtle logical inconsistencies occur around Np = 1 = rad, equivalent to say 1 s = $0$ or i = 1.  I can see one logically consistent way forward, though this will involve a retrospective formal reinterpretation of the dB as applied to a ratio of field quantities to apply to an associated power quantity.  —Quondum 15:58, 27 October 2019 (UTC)


 * I don't see how rad and Np can have different dimensions, because they are the units of the real and imaginary parts of the same (complex) phase. But I see no reason why Np and dB need to have the same dimensions. I'm glad it's not my job though. Persuading Kim Yong Un to disarm is a cushy job by comparison. Dondervogel 2 (talk) 17:36, 27 October 2019 (UTC)
 * Yep, I'm getting myself a little muddled. One logically has that Np = rad in the current formulation.  My other point remains: that neither of these units should be equated to a real number.  You may be correct about the difficulty of persuading a committee.  It needs someone as analytical as Peter Mohr and a few decades to spare.  The SI at least seems to be acknowledging that angle being dimensionless is not the only way in its choice of words: "Plane and solid angles, when expressed in radians and steradians respectively, are in effect also treated within the SI as quantities with the unit one".  —Quondum 22:12, 27 October 2019 (UTC)
 * What is the logic by which the real and imaginary parts of a complex logarithm are of the same unit, unless non-dimensional? Dicklyon (talk) 02:22, 28 October 2019 (UTC)
 * In the current formulation, yes, the natural exponential function is used, and the real an imaginary parts are inherently non-dimensional. However, the choice of the base is in fact free, and in a reformulation, a base-agnostic approach can be used that requires dimensional units, which I think would be a real benefit to metrology.  The relationship between the units of the "real" and "imaginary" parts can also be broken in a reformulation to reflect our intuitive separation of these units.  —Quondum 14:21, 28 October 2019 (UTC)

Suggestion: draft a "list of pitfalls when trying to explain level quantities in Wikipedia," then submit it to the ISO committee as feedback (and/or to the WikiJournal of Science for publication). fgnievinski (talk) 16:36, 27 December 2019 (UTC)