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Supplemental note on inharmonic discrete equations (rev.3.4)

Note (added on June 18, 2011): This supplemental note was originally written to clarify the mistake by the IP user 70.109.187.11 at 04:49 Match 8, 2011 (UST). On revision 417732066, this IP user had changed the description of variable φk[n] from "phase" into "instantaneous phase", but it was clearly confused phase φ with instantaneous phase θ, thus, already fixed on the latest version of article after the long discussions on this talk page.

 Revision as of 04:33, 8 March 2011 70.109.187.11 (→‎Theory: Changed k_max to K and \varphi to \phi. Added a couple of wikilinks.) Revision as of 04:49, 8 March 2011 70.109.187.11 (→‎Theory: A little more wikilinking.) : ''φk''[''n''] is the phase of the ''k''-th harmonic at discrete time ''n'', : ''φk''[''n''] is the [[instantaneous phase]] function of the ''k''-th harmonic at discrete time ''n'',

However, IP user 70.109.xxx.xxx (probably same user) didn't yet admit this mistake was originally caused by 70.109.187.11, and even claims as if it was caused by others. (did another user use 70.109.xxx.xxx on this article ?)

I'm still expecting his sincere confession about his own mistake on March 6, 2011. sincerely, --Clusternote (talk) 03:00, 18 June 2012 (UTC) [mod] --Clusternote (talk) 04:05, 18 June 2012 (UTC)

Revision History

[rev.3]: With the great help of participants of related discussion, I've revised above note as following. --Clusternote (talk) 16:42, 19 January 2012 (UTC) [added summary]--Clusternote (talk) 22:29, 9 March 2012 (UTC)

[rev.3.2]: Added supplements for more about instantaneous terms, mention on Rodet & Depalle 1992, and introduction of multiple band-limited (I'll add more on the later). --Clusternote (talk) 23:26, 26 January 2012 (UTC)

[rev.3.3]: cleanup. moved complicated part into footnote, etc.

[rev.3.4]: changed leading paragraph of "Introduction of instantaneous terms"; it can be explained as an extension of static frequency into instantaneous frequency by taking the limit as ${\displaystyle \Delta {t}\rightarrow 0\ }$.

Supplemental note using continuous form — for section Inharmonic form of Discrete-time equations

The sub-section "Inharmonic form" was described with discrete equations using following notions:

• "instantaneous phase" ${\displaystyle \theta \ }$ , instead of "phase offset"  ${\displaystyle \phi \ }$
(possibly reverse use of these characters might be also familiar),
• "instantaneous frequency"  ${\displaystyle {\frac {1}{n}}\sum _{i=0}^{n}f_{k}[i]\ }$  in discrete form (also, divided by discrete time ${\displaystyle n\ }$),[note 1]
instead of "time-varying frequency"  ${\displaystyle f_{k}[i]\ }$ .
For readers not familiar with above, descriptions using these are possibly not easy to understand at a glance. The following is a try to provide supplemental explanations for these readers.

A wave in continuous form (correspond to a partial of the 2nd equation)

A wave is expressed with angular frequency ${\displaystyle \omega _{k}\ }$ and phase ${\displaystyle \phi _{k}(t)\ }$ , as following:

${\displaystyle x_{k}(t)\ =r_{k}(t)\cdot \cos(\omega _{k}\cdot {t}+\phi _{k}(t))\ \ \ \ \ \ \ \ (\omega _{k}>0,\ {\mbox{real signal}})}$

(1)

Introduction of instantaneous terms [CHANGED]
As a simple case, the static frequency is defined as the number of waves per unit time, as following:
${\displaystyle f=\omega /2\pi ={\frac {(\omega \cdot (t+\Delta {t})+\phi )-(\omega \cdot {t}+\phi )}{2\pi \Delta {t}}}\ \ \ \ {\mbox{(average over a unit time)}}}$
On the other hand, if we need practical additive synthesis beyond the toy program, we should also consider the time-varying frequency and time-varying phase (probably passed from analysis part of analysis/resynthesis type additive synthesis system including several speech synthesis system). On that case, above static frequency definition is not useful because it gives merely the average frequency over a unit time. Instead, for the time-varying frequency, we should extend static frequency into instantaneous frequency of each timing by taking the limit as ${\displaystyle \Delta {t}\rightarrow 0\ }$ , as roughly following:
{\displaystyle {\begin{aligned}\lim _{\Delta {t}\rightarrow 0}f&=\lim _{\Delta {t}\rightarrow 0}{\frac {(\omega (t+\Delta {t})\cdot (t+\Delta {t})+\phi (t+\Delta {t}))-(\omega (t)\cdot {t}+\phi (t))}{2\pi \Delta {t}}}\\&={\frac {d(\omega (t)\cdot {t}+\phi (t))}{2\pi dt}}\ \ \ \ \ \ \ \ {\mbox{(derivative of instantaneous phase'')}}\end{aligned}}}
On the rest of this subsection, we explain the details of these instantaneous terms.
Let's follow the style of signal analysis !
Following the style of signal processing, above real-valued signal can be extended into a complex form,[note 2][note 3][note 4]called ...
Analytic representation (analytic signal)
{\displaystyle {\begin{aligned}x_{a}(t)&=r_{k}(t)\cdot e^{j(\omega _{k}t+\phi _{k}(t))}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mbox{(complex signal)}}\end{aligned}}}

Instantaneous phase  ${\displaystyle \theta _{k}\ }$  is given by argument function.[note 5] On the above case, it is given as:

{\displaystyle {\begin{aligned}\theta _{k}(t)&={\mbox{arg}}(x_{a}(t))=\omega _{k}\cdot {t}+\phi _{k}(t)\ \ \ \ \ \ ({\mbox{where}}\ \theta _{k}(0)=\phi _{k}(0)\ {\mbox{for}}\ t=0)\end{aligned}}}

(4)

Instantaneous frequency  ${\displaystyle f_{k}(t)\ }$  is defined by differentiation of above instantaneous phase.
If phase ${\displaystyle \phi _{k}(t)\ }$  is time-invariant, it is ignorable, and whole expression is:

${\displaystyle f_{k}(t)=\omega _{k}(t)/2\pi ={\frac {d}{dt}}\theta _{k}(t)/2\pi }$

(5)

Redefinition of instantaneous phase ${\displaystyle \theta _{k}(t)\ }$ is introduced from above, using angular frequency ${\displaystyle \omega _{k}(t)\ }$, as:

${\displaystyle \theta _{k}(t)=\int _{-\infty }^{t}\omega _{k}(\tau )d\tau =\int _{0}^{t}\omega _{k}(\tau )d\tau +\theta (0)\ \ \ \ (\omega _{k}>0)}$

(6)

On discretization, above integral form can be rewritten into its discrete form, using substitutions:
${\displaystyle t\rightarrow n/f_{s}=nT\ }$    and    ${\displaystyle \omega _{k}(t)\rightarrow \omega _{k}[n]=2\pi f_{k}[n]\ }$
as following:

{\displaystyle {\begin{aligned}\theta _{k}[n]&=\sum _{i=-\infty }^{n}\omega _{k}[i]=\sum _{i=0}^{n}\omega _{k}[i]+\theta _{k}[0]\\&=2\pi T\sum _{i=0}^{n}f_{k}[i]+\theta _{k}[0]\ \ \ \ \ \ ({\mbox{where}}\ \theta _{k}[0]=\phi _{k}[0]\ {{\mbox{for}}\ n=0})\end{aligned}}}

(7)

By differentiating ${\displaystyle \theta _{k}[n]\ }$, above is expressed as:

{\displaystyle {\begin{aligned}\theta _{k}[n]&=\theta _{k}[n-1]+2\pi Tf_{k}[n]\end{aligned}}}

(8)

Comparison with expressions shown on other sources
• Almost same expressions are seen on Smith III 2011 [5][cite 1] in the continuous form, and on Smith III & Serra 2005 [6].[cite 2][note 6]
• [ADDED] Similary, same equation forms were also seen on Rodet & Depalle 1992, p. 2nd page. On their expression, above instantaneous phase ${\displaystyle \theta \ }$ was replaced to phase offset ${\displaystyle \Phi \ }$, and above initial (or static) phase offset ${\displaystyle \phi _{k}[0]\ }$ was not apparent. [note 7] Thus, as far as above equations are concerned, at least that paper seems slightly hard to be called reliable source.

Introduction of band-limited time-varying terms

According to Papoulis 1977, p. 184,
"We shall say that a function ${\displaystyle f(t)\ }$ is bandlimited if its Fourier transform is zero outside a finite interval ( ${\displaystyle F(\omega )=0\ }$    ${\displaystyle {\mbox{for}}|\omega |>\sigma \ }$ ) and its energy ${\displaystyle E\ }$ is finite."
(Supplement: On the above quotation, ${\displaystyle \sigma \ }$ means bandwidth, and it seems often denoted by the fundamental frequency ${\displaystyle \omega _{0}=2\pi f_{0}\ }$, as shown on below[cite 3])
For the time-varying terms other than frequency (i.e. phase and amplitude ), if these were band-limited below the fundamental frequency ${\displaystyle f_{0}\ }$,[cite 3][cite 4] whole above discussion is almost applicable, with a few modifications. (The implementation details of band-limited terms are not mentioned on this note)

For time-varying phase ${\displaystyle \phi _{k}(t)\ }$ on equation 4

If ${\displaystyle \phi _{k}[n]\ }$ is band-limited as above,[cite 3] whole above discussion is almost applicable with a little modifications—subsequent equations (eq. 5, 6, 7, 8) should regard on the phase ${\displaystyle \phi _{k}(t)\ }$, as following:

{\displaystyle {\begin{aligned}f_{k}(t)&=\omega _{k}(t)/2\pi ={\frac {d}{dt}}\left(\theta _{k}(t)-\phi _{k}(t)\right)/2\pi \end{aligned}}}

(5-2)

{\displaystyle {\begin{aligned}\theta _{k}(t)&=\int _{-\infty }^{t}\omega _{k}d\tau +\phi _{k}(t)=\int _{0}^{t}\omega _{k}d\tau +\theta (0)+\phi _{k}(t)\ \ \ \ (\omega _{k}>0)\\\end{aligned}}}

(6-2)

{\displaystyle {\begin{aligned}\theta _{k}[n]&=\sum _{i=-\infty }^{n}\omega _{k}[i]+\phi _{k}[n]=2\pi T\sum _{i=-\infty }^{n}f_{k}[i]+\phi _{k}[n]=2\pi T\sum _{i=0}^{n}f_{k}[i]+\theta _{k}[0]+\phi _{k}[n]\end{aligned}}}

(7-2)

{\displaystyle {\begin{aligned}\theta _{k}[n]&=\theta _{k}[n-1]+2\pi Tf_{k}[n]+\phi _{k}[n]-\phi _{k}[n-1]\end{aligned}}}

(8-2)

Equation 7-2 requires  ${\displaystyle \phi _{k}[0]=0\ }$ , because  ${\displaystyle \theta _{k}[0]=\theta _{k}[0]+\phi _{k}[0]\ \ {\mbox{for}}\ n=0\ }$.
(Note: I'm expecting more accurate, precise discussions might be found in somewhere around DFT, STFT, or these application, Phase vocoder)

For time-varying amplitude ${\displaystyle r_{k}[n]\ }$

If ${\displaystyle r_{k}[n]\ }$ is band-limited as above,[cite 4] whole above discussion is directly applicable without modification.

Multiple use of time-varying terms [ADDED]

Multiple use of band-limited time-varying terms may be not always band-limited even if each term was individually band-limited on the single time-varying model, because, on the estimated bandwidths for each modulation type (see below), several sum-rules between bandwidths (or merely a rule of thumb) are naturaly expected (in other words: when each time-invariant term was replaced by time-varying version, total bandwidth may be mostly widen rather than unchanged or narrowed).
AM bandwidth   ${\displaystyle \ \approx \ 2f_{m}\ }$
FM bandwidth   ${\displaystyle \ \approx \ 2(\Delta {}f+f_{m})\ }$
PM bandwidth   ${\displaystyle \ \approx \ 2(\Delta \theta +1)f_{m}\ }$
where ${\displaystyle f_{m}}$  is frequency of simplified modulation signal ${\displaystyle x_{m}(t)=A_{m}\cos(2\pi f_{m}t+\phi _{m})}$
I'll add details on the later.

General expression of inharmonic additive synthesis
[EXTENDED]
Using above equation 7-2, inharmonic additive synthesis is expressed as:

{\displaystyle {\begin{aligned}y[n]&=\sum _{k=1}^{K}r_{k}[n]\cos \left(\theta _{k}[n]\right)\\&=\sum _{k=1}^{K}r_{k}[n]\cos \left(2\pi T\sum _{i=-\infty }^{n}f_{k}[i]+\phi _{k}[n]\right)\\&=\sum _{k=1}^{K}r_{k}[n]\cos \left(2\pi T\sum _{i=0}^{n}f_{k}[i]+\theta _{k}[0]+\phi _{k}[n]\right)\\\end{aligned}}}

(10)

Notes

1. ^ As explained on the later section, instantaneous frequency can be defined as derivative of instantaneous phase  ${\displaystyle \theta _{k}(t)\ }$  as:  ${\displaystyle {\frac {d}{dt}}\theta _{k}(t)/2\pi }$  in continuous form.
2. ^ [MOVED TO NOTE] Following the style of signal processing, above real-valued signal can be extended into a complex form, called "analytic representation":
{\displaystyle {\begin{aligned}x_{a}(t)&=x_{k}(t)+j\cdot {\tilde {x}}_{k}(t)\\\end{aligned}}}
where ${\displaystyle j\ }$ denotes imaginary unit. Its real part is same as above ${\displaystyle x_{k}(t)\ }$ , and additional imaginary part ${\displaystyle {\tilde {x}}_{k}(t)\ }$  given by Hilbert transform of ${\displaystyle x_{k}(t)\ }$, is known to be expressed as:

${\displaystyle {\tilde {x}}_{k}(t)=H(x)(t)=r_{k}(t)\sin(\omega _{k}\cdot {t}+\phi _{k}(t))\,}$

(2)

Analytic representation (analytic signal) is expressed as:

{\displaystyle {\begin{aligned}x_{a}(t)&=r_{k}(t)\left[\cos(\omega _{k}\cdot {t}+\phi _{k}(t))+j\cdot \sin(\omega _{k}\cdot {t}+\phi _{k}(t))\right]\\&=r_{k}(t)\cdot e^{j(\omega _{k}t+\phi _{k}(t))}\ \ \ \ {\mbox{(complex signal)}}\end{aligned}}}

(3)

3. ^ On this note, Hilbert transform of ${\displaystyle x_{k}(t)\ }$  is denoted by ${\displaystyle {\tilde {x}}_{k}(t)\ }$ , instead of ${\displaystyle {\hat {x}}_{k}(t)\ }$ .
reason: the later notation is also often used to denote analytic representation itself  ${\displaystyle x_{a}(t)=x_{k}(t)+H(x_{k})(t)\ }$ , or even Fourier transform ${\displaystyle {\mathcal {F}}(x_{k})(t)\ }$  of ${\displaystyle x_{k}(t)\ }$. To avoid confusion, the former notation is more appropriate on here.
4. ^ Hilbert transform of ${\displaystyle \cos \ }$ and ${\displaystyle \sin \ }$ functions are known to given by ${\displaystyle \pi /2\ }$ phase delayed signal. Although detail is omitted on here, it is led by the convolution of ${\displaystyle x_{k}(t)\ }$  with the function ${\displaystyle h(t)=1/(\pi t)\ }$ in the form of principal value integral, as following:
${\displaystyle H(x_{k})(t)={\frac {1}{\pi }}\ p.v.\int _{-\infty }^{\infty }{\frac {x_{k}(\tau )}{t-\tau }}d\tau \ \ \ \ {\mbox{or}}\ \ \ \ H(x_{k})(t)=-{\frac {1}{\pi }}\lim _{\epsilon \downarrow 0}\int _{\epsilon }^{\infty }{\frac {x_{k}(t+\tau )-x_{k}(t-\tau )}{\tau }}d\tau }$
where ${\displaystyle p.v.\ }$ denotes Cauchy principal value.
5. ^
${\displaystyle {\mbox{arg}}(x_{a}(t))=\arctan \left({\frac {\Im (x_{a}(t))}{\Re (x_{a}(t))}}\right)\mod \pi \ \ \ \ {\mbox{for}}\ x_{a}(t)\neq 0}$
6. ^ [ADDED] According to Smith III 2011 [1] and Smith III & Serra 2005 [2]
${\displaystyle \theta _{k}(t)=\int _{0}^{t}\omega _{k}(\tau )d\tau +\phi _{k}(0)\ \ {\xrightarrow[{}]{discreptize}}\ \ \,\theta _{k}[n]=2\pi T\sum _{i=0}^{n}f_{k}[i]+\phi _{k}[0]}$
${\displaystyle {\hat {\Theta }}_{k}(n)\triangleq {\hat {\Theta }}_{k}(n-1)+2\pi T{\hat {F}}_{k}(n)}$
(Note: on above expressions, several variable names are replaced as: ${\displaystyle i\rightarrow k\ }$ , integration variable ${\displaystyle t\rightarrow \tau \ }$ .)

where [3] [4]

{\displaystyle {\begin{aligned}{\tilde {x}}'_{m}(e^{j\cdot \omega _{k}})&:\end{aligned}}}  STFT of ${\displaystyle x_{m}(n)\triangleq x(n-mR)\ }$ , at ${\displaystyle k}$th bin, ${\displaystyle m}$th frame.
${\displaystyle R\ }$ is hop size (hops) of STFT,
"${\displaystyle {\tilde {}}\ }$" (over tilde) denotes applying spectral analysis window ${\displaystyle w(n)\ }$ , and
"${\displaystyle '\ }$" (prime) denotes zero-padding of both side on FFT frame.
{\displaystyle {\begin{aligned}\Theta _{k}(m)&\triangleq \angle {\tilde {x}}'_{m}(e^{j\cdot \omega _{k}})\ \ \ \ [{\mbox{radians}}]\\F_{k}(m)&\triangleq {\frac {\Theta _{k}(m)-\Theta _{k}(m-1)}{2\pi RT}}\ \ \ \ [{\mbox{Hz}}]\\T&=1/f_{s}\ \ \ \ [{\mbox{seconds}}]\\\end{aligned}}}
7. ^ [ADDED] On Rodet & Depalle 1992, "Theory and the oscillator method" section, they used following equations:
{\displaystyle {\begin{aligned}s[n]&=\sum _{k=1}^{K}c_{k}[n]\\c_{k}[n]&=a_{k}[n]\cdot \cos(\Phi _{k}[n])\\\Phi _{k}[n]&=\Phi _{k}[n-1]+{\frac {2\pi }{Sr}}f_{k}[n]\end{aligned}}}
(Note: on above expressions, several variable names are replaced as: ${\displaystyle j\rightarrow k\ }$ ,   ${\displaystyle J\rightarrow K\ }$ .)

where each term of ${\displaystyle k}$th partial at sample time ${\displaystyle n}$ had been described as

${\displaystyle f_{k}[n]\ }$ for frequency,
${\displaystyle a_{k}[n]\ }$ for amplitude,
${\displaystyle \Phi _{k}[n]\ }$ for phase (Note: probably typo of the instantaneous phase).

References

1. ^
2. ^
3. ^ a b c Kwakemaak & Sivan 1991, p. 613–614
4. ^ a b Papoulis 1977, p. 121

I expect gentle criticism : ) --Clusternote (talk) 14:02, 19 January 2012 (UTC) [rev.3.2]Clusternote (talk) 23:26, 26 January 2012 (UTC) [added summary on rev.3.3]--Clusternote (talk) 22:29, 9 March 2012 (UTC) [rev.3.4]Clusternote (talk) 00:51, 10 April 2012 (UTC)

Definition and scope

Google Books can help arrive at a neutral definition of additive synthesis. I have studied the first 20 results of an anonymous Google Books search of "additive synthesis" -inauthor:"Books, LLC" carried out today with the preview and full view option enabled.

• 19 (95 %) discuss sound synthesis,
• 1 (5 %) discusses light.

Of the 19 that discuss sound synthesis

• 12 (63 %) only discuss additive synthesis as having sinusoidal components,
• 3 (16 %) define additive synthesis as having sinusoidal components and discuss non-sinusoidal extensions and variants separately, not including them in the definition of additive synthesis,
• 2 (11 %) initially define additive synthesis as having sinusoidal components and give a second definition with added noise,
• 1 (5 %) defines additive synthesis as having sinusoidal and noise components,
• 1 (5 %) defines additive synthesis as any sum of components.

I recommend a compromise organization for the article similar to the second group (16 %), a separation between non-sinusoidal extensions and variants from a main sinusoidal definition. In my analysis of the Google Books results, I did not differentiate between harmonic and non-harmonic sinusoids or between amplitude- and frequency-modulated sinusoids and non-modulated sinusoids. Olli Niemitalo (talk) 11:50, 2 February 2012 (UTC)

I agree with separation between non-sinusoidal extensions and variants, however, I disagree with treating sinusoidal definition as main part. Although sinusoidal had been historically main stem of the genre, however, theoretical universality of it (such as "theoretically all quasi-periodical waveform is re-produced by ...") became already out of date due to disregard of "transient waves" and possibly "non-linear characteristic of human ears".
Now, we should select more generic, extensible definition as basis of this synthesis family, to more reasonably categorize newer, practical synthesis emerged after 1980s. The term "Additive synthesis" is not only for the classical sinusoidal sound synthesis before the late 1970s.
BTW: I think that the statistics on the books for generic readers seems not so reliable as the basis of more specific article's scope, because that type of statistics tends to be biased with introductory article for beginners which preferred ease of understand for beginners, than relevance of definition inside the specific field.   ( For example, even well known MIT's The CSound Book didn't supply enough information on advanced additive synthesis other than unpractical toy program using sinusoidal. This type of toy program for beginner is almost ignorable. Otherwise, almost all the special article on Wikipedia should be redefined by the ambiguous beginner's book, instead of more relevant reliable sources in special field. (Supplement for honor of this book, phase vocoder section is possibly technically interesting, along with historical software design of CSound itself)) --Clusternote (talk) 12:17, 2 February 2012 (UTC)
P.S. I think that we need several advices or second opinion from external researchers / developers / musicians to avoid intolerant local rules. --Clusternote (talk) 13:44, 2 February 2012 (UTC)

I also did a similar analysis of results returned today by Google Scholar from a search "additive synthesis" on works published 1990—2012, limiting the analyzed results to those that 1) were available in full text at the library of my university, 2) were concerned with sound, and 3) were among the 20 first results obtained that passed the previous two criteria. Of the 20 results

• 13 (65 %) define or cite a work with a definition of additive synthesis as having sinusoidal components only,
• 1 (5 %) define additive synthesis as having sinusoidal components and discuss non-sinusoidal extensions and variants separately, not including them in the definition of additive synthesis,
• 2 (10 %) are unclear in whether added noise components are external or included by definition in additive synthesis,
• 4 (20 %) define additive synthesis as having both sinusoidal and noise components.

Olli Niemitalo (talk) 16:54, 2 February 2012 (UTC)

Thanks for your statistics. I couldn't imagine how fast your work is. However, in my eyes, it still seems unrealistic. If you want to make reliable statistics, at least following points concerning the validity of statistical population and statistically sampling, should be always explicitly explained::
a) What is search criteria and sorting criteria ?                     [statistical population]
Providing the search URL would be generally useful way for this purpose.
However sorting criteria of Google Scholar seems slightly non standard or unclear (probably it use a variation of PageRank instead of traditional citation count).
b) Where is the sampled paper list for statistics?                 [statistically sampling]
In general the statistics is, an numerical abstraction utilizing sampling to show the tendency of certain aspects of statistical population consisting of more complex things. In the viewpoint of verification, as possible as sampled paper list should be provided.
Also, your condition 2 "concerned with sound" is superficially seems to be natural, however it is biased; as I already slightly mentioned, several scientific musician/media artists in academic sites and researchers on computer music tend to carry out their research as speech synthesis due to limitation of funding and due to convenience of research trends.
For the neutral point of view, we should carefully verify the intentions of each researches. It seems slightly hard thing for quick statistics, however, we need not hasty not enough conclusion. Also I want to try to statistics, on the later. --Clusternote (talk) 23:51, 2 February 2012 (UTC)
At this point, I can only ask you to please do your own statistics. I've done enough. Olli Niemitalo (talk) 07:50, 3 February 2012 (UTC)
No, your statistics is yet incomplete because lacking two essential information indispensably required for every reliable statistics mentioned on above. On the reliable science and engineering, reliable statistics is one of most basic skill for all from freshman of university to researcher, engineer.
I'm very sad to know you haven't yet enough skill on statistics which is even required for student's experiments report. --Clusternote (talk) 09:04, 3 February 2012 (UTC)
Well, Olli, we are "intolerant locals", we lack "enough skill on statistics which is even required for student's experiments report", we're "confused", we use "uncertain English", we're "uncooperative, offensive", "ill-mannered". We lack "more practical knowledge beyond the classical toy program". I, evidently do not understand the "essential equations on supplemental note were too complicated, very sad to say, probably [I] didn't enough studied undergraduate mathematics underlying these." We are so incompetent, and when Clusternote demonstrates his superior knowledge and competence, we resort to personal attacks. And we've never invited Clusternote to contribute to the article in meaningful ways. 71.169.185.12 (talk) 18:08, 8 February 2012 (UTC)

I've requested for a third opinion on the status of the sinusoidal definition as the main definition in the article, from some of the editors who have contributed to this article before and from Wikipedia:WikiProject Linguistics, Wikipedia talk:WikiProject Professional sound production, Wikipedia talk:WikiProject Musical Instruments, and Wikipedia talk:WikiProject Electronics‎. Olli Niemitalo (talk) 11:20, 3 February 2012 (UTC)

I picked this up from Wikiproject Electronics and am uninvolved in this article. I find the statistics provided by Olli Niemitalo to be acceptable for the purposes of establishing the common definition of the term in reliable sources. For the purposes of an encyclopaedia entry, it is not necessary to have a statistical analysis that could be defended as a PhD thesis. The claim that there are flaws in the methodology is not a very convincing argument for rejecting the results without alternative statistics to a different methodology or a plausible explication of how the flaws have affected the result. As always, we should write from the sources available. My instinct on this is that Wikipedia articles should start from the simple cases and build from there. Perhaps the lede and introduction could compromise with such phrases as "normally sinusiodal", or "in its simplest form sinusoidal" in order not to rule out other elements being used to construct a synthesis. As an aside, it is much easier for cheap musical instruments to achieve synthesis with non-sinusoids such as square waves and pulses since these can be digitally generated. Such techniques for mass-market manufacturing should not be entirely ignored by the article. SpinningSpark 11:40, 3 February 2012 (UTC)
I would like to avoid definitions like "in its simplest form sinusoidal", because the most common definition refers exclusively to synthesis from (possibly time-varying) sinusoidal components. Something like "normally sinusoidal" sounds acceptable, because it can be understood to reflect that. (Edit 4 Feb 2012: but it may not be acceptable by Wikipedia policy) However, this leaves open the question whether in the body of the article we always need to specify additive synthesis to be with sinusoidal components when we mean that or can we write simply additive synthesis. The latter use, which I find more natural, could be disambiguated early on in the article with something like "This article adopts the above sinusoidal definition for additive synthesis. However, some authors define additive synthesis as .... Methods falling under the broader definitions are discussed in sections...". Unfortunately I have not found a single source with an assessment of the apparent conflict of definitions by different authors. Olli Niemitalo (talk) 13:04, 3 February 2012 (UTC)
Olli, why you can't show searching URL and list of sampled papers ? If you really do it, you can quickly show these.  Statistics lacking verifiability should be rejected. --Clusternote (talk) 12:35, 3 February 2012 (UTC)
Clusternote, Olli's definitions confirm what we have been discussing for months. I don't think there is any new information here. I think the burden of proof is on you to demonstrate some contradictory evidence as Spinningspark has already said. Ross bencina (talk) 13:31, 3 February 2012 (UTC)
Ross, on my last post, I merely request completion of his incomplete statistics. It is purely technical issue.
Note: On our fields (physics and informatics), this type of defeats is sometimes called "Voodoo statistics" because it is not verifiable thus not reliable. It is often seen on advertisement for impression operation, unreliable research reports, or several overlapping fields lacking proper statistics skill.--Clusternote (talk) 02:47, 4 February 2012 (UTC)
My \$0.25... I think Olli is on the right track, surveying sources. But I suggest avoiding the statistics or "simplest" debate by using words like "some authors" and "other authors" for each claimed definition. Wording like that is non-POV and verifiable if multiple cites are added to each position. Include page numbers and quotes (in the ref, not the text). "Simplest" starts to sound POV -- avoid. "Most common" may seed a statistics debate in the future -- avoid. Remember a line from WP:Verifiability: The threshold for inclusion in Wikipedia is verifiability, not truth—whether readers can check that material in Wikipedia has already been published by a reliable source, not whether editors think unsourced material is true. --Ds13 (talk) 17:54, 3 February 2012 (UTC)
Yes, that is Wikipedia policy; we cannot make statements in the article based on unsourced estimates of the relative proportions of different definitions in reliable sources. But it is also Wikipedia policy that (quoting Wikipedia:Verifiability#Neutrality): "All articles must adhere to the Neutral point of view policy (NPOV), fairly representing all majority and significant-minority viewpoints published by reliable sources, in rough proportion to the prominence of each view." To figure out the suitable weighting (see Wikipedia:Neutral point of view#Due_and_undue_weight) for the article, unsourced estimates (such as the presented statistics) of relative proportions in reliable sources must be acceptable. Olli Niemitalo (talk) 19:27, 3 February 2012 (UTC)
Agree completely, as long as the unsourced estimates are the stuff of talk pages. (i.e. I wouldn't expect to see an original or unsourced "two thirds of experts believe" in the article... and I doubt that's what you were proposing anyways) If a vast majority of experts believe that additive synthesis is X, then it will be easy for the reader to confirm this fact through lots of cites, and then I think it can be stated that way. I'm thinking ahead, when the current editors of this overhaul perhaps no longer watch over the article... a future editor might reasonably come along and see something like "majority" or "most common" and remove or neuter it. But that will happen no matter what, so if in doubt, be bold and see what happens! Thanks for listening and thank you all for putting serious effort into the article recently. --Ds13 (talk) 20:36, 3 February 2012 (UTC)
I feel that the Additive Synthesis article should be aimed almost entirely at the addition of sinusoids. This is what in my experience is usually meant by 'additive synthesis', and this view is backed up by Olli's searches. Possibly, in the absence of a separate article on tones made by the addition of non-sinusoidal waveforms, the 'Broader definitions' section should be expanded slightly, but it needs to be made clear that such synthesis with non-sinusoidal waveforms is usually considered an extension to additive synthesis, rather than a variant of it. Chrisjohnson (talk) 18:05, 3 February 2012 (UTC)

MizsaBot doesn't like your "Supplemental note on inharmonic discrete equations (rev.3.3)", Cluster.

Don't talk back again and frustrate her efforts, Cluster, or she'll slap you up big time. 71.169.185.196 (talk) 05:33, 20 March 2012 (UTC)

I'm not interested in 71.169.x.x's talk at all

I'm not interested in 71.169.x.x's talk at all. --Clusternote (talk) 08:31, 20 March 2012 (UTC)

Note how Olli fixed your tagging problem with Mizsa.
Clusternote, I was serious about that you were right about the lacking in the article of a section on analysis/resynthesis and the suggestion (I don't presume to issue "invitations") for you to do something about it, including your graphic that was based on Beauchamp/Horner work. That would be a valuable addition to the article. 71.169.185.196 (talk) 20:58, 20 March 2012 (UTC)
That there is no article about McAulay-Quatieri sinusoidal model or any MQ anything except this article and Nellymoser (WTF is that?!!). And the sinusoidal model is pathetic. It's like Dimensional analysis without the Buckingham π theorem. Even worse. So, besides this article, there needs to be a good Sinusoidal modeling article. And maybe this could point to that as the "Main article", but not as it is now.
So it looks like there could be substantive improvements to the Instantaneous frequency and, essentially, a new article at Sinusoidal modeling possibly replacing Sinusoidal model. Dunno if anyone else wants to fix any of that. I know there is an EE culture hanging out at Instantaneous phase but I think they can be persuaded that the article can be made much better. Maybe I'll make another stab at one or both of these, but I don't think that I have all of the necessary ideas about it. 71.169.185.196 (talk) 22:59, 20 March 2012 (UTC)

Bold text

Should Korg Wavestation and Vocaloid be included here as products employing Additive synthesis?

I'm about to yank it out again, but since Clusternote is persistent and apparently insistent, we should discuss this. I am prepared to yank it again, if there is no support for the addition. As far as I can see, neither of these products employ additive synthesis. 70.109.185.7 (talk) 03:45, 23 May 2014 (UTC)

On the first half:   The 1st image is merely diverted to show how timbre-frame crossfading of Synclavier is done; on this context, Korg Wavestation is not related. If more appropriate image was exist, current image should be replaced.
On the second half:   The 2nd image (in Germany) is shown to describe more sophisticated method used on Vocaloid; on which peaks of harmonic partials are morphed between timbre frames. Also, Vocaloid is an advanced application of additive analysis/re-synthesis technology because its spectral voice model is extension of Spectral Modeling Synthesis (SMS) based on additive analysis/re-synthesis.
--Clusternote (talk)
Neither images have anything to do with Additive synthesis. 65.183.156.110 (talk) 02:19, 26 March 2015 (UTC)
I've already replied on your question in 24 May 2014. Stop stalking immediately. --Clusternote (talk) 02:46, 26 March 2015 (UTC)

I just deleted the section suggesting that wavetable synthesis was a poor man's additive synthesis. It is nothing of the sort. The awful paper it cites as an authority is saying nothing more than if you take multiple oscillators and sum them, perhaps with separate envelopes, that's "additive synthesis": it simply does so in a wavetable context, and furthermore makes the claim that wavetables may be viewed as "preprocessed" sums of sine waves. This is trivialization of additive synthesis to the point where it loses all meaning. Under this definition, any analog subtractive synthesizer with two oscillators is "additive" -- indeed, any polyphonic synthesizer is "additive".

Additive synthesis needs a more specific definition than this: otherwise everything is additive if it involves producing N>2 voices. — Preceding unsigned comment added by 151.35.82.126 (talk) 09:08, 3 April 2017 (UTC)

Uhm, sorry but you're wrong. If all of the overtones are strictly harmonic, wavetable can do anything additive can do. But wavetable fails if there are overtones of significant amplitude that are also sufficiently detuned from their harmonic value (like bells and such). I have reverted the deletion and added another cite to an AES paper from Andrew Horner that says essentially the same thing. Also, Group Additive Synthesis uses wavetable synthesis for each harmonic group. Group additive doesn't really have any point to it without wavetable being the basic engine. That's why group additive should be a subsection under wavetable. 173.48.64.110 (talk) 00:19, 27 June 2017 (UTC)

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