Editing Organ flue pipe scaling (section)

==Töpfer's ''Normalmensur'' ==

The system most commonly used to fully document and describe scaling was devised by [[Johann Gottlob Töpfer]].<ref>{{cite book |last1=Adelung |first1=Wolfgang |title=Einführung in der Orgelbau |date=1991 |publisher=Breitkopf & Härtel |isbn=3-7651-0279-2 |pages=85–86}}</ref> Since varying the diameter of a pipe in direct proportion to its length (which means it varies by a factor of 1:2 per octave) caused the pipes to narrow too rapidly, and keeping the diameter constant (a factor of 1:1 per octave) was too little, the correct change in scale must be between these values. Töpfer reasoned that the cross-sectional area of the pipe was the critical factor, and he chose to vary this by the [[geometric mean]] of the ratios 1:2 and 1:4 per octave. This meant that the cross-sectional area varied as <math>1 : \sqrt{8}</math>. In consequence, the diameter of the pipe halved after 16 semitone intervals, i.e. on the 17th note (musicians count the starting-note as the first, so if C is the first note, C# is the second, differing by one semitone). Töpfer was able to confirm that if the diameter of the pipes in a rank halved on the 17th note, its volume and timbre remained adequately constant across the entire organ keyboard. He established this as a standard scale, or in German, '''Normalmensur''', with the additional stipulation that the internal diameter be {{Convert|155.5|mm|abbr=on}} at 8&prime; C (the lowest note of the modern organ compass) and the mouth width one-quarter of the circumference of such a pipe.

Töpfer's system provides a reference scale, from which the scale of other pipe ranks can be described by means of ''[[semitone|half-tone]]'' deviations larger or smaller (indicated by the abbreviation ''ht''). A rank that also halves in diameter at the 17th note but is somewhat wider could be described as "+ 2 ht" meaning that the pipe corresponding to the note "D" has the width expected for a pipe of the note "C", two semitones below (and therefore two semitone intervals wider). If a rank does not halve exactly at the 17th note, then its relationship to the Normalmensur will vary across the keyboard. The system can therefore be used to produce Normalmensur variation tables or line graphs for the analysis of existing ranks or the design of new ranks.

The following is a list of representative 8&prime; stops in order of increasing diameter (and, therefore, of increasingly fundamental tone) at [[middle C]] with respect to Normalmensur, which is listed in the middle. Deviations from Normalmensur are provided after the pipe measurement in brackets.

* Viole d'orchestre (thin, mordant string stop): 35.6&nbsp;mm [-10 ht]
* Salicional (broader-toned, non-imitative string stop): 40.6&nbsp;mm [-7 ht]
* Violin diapason (thin-toned principal stop): 46.2&nbsp;mm [-4 ht]
* Principal (typical mid-scale principal stop): 50.4&nbsp;mm [-2 ht]
* Normalmensur: 54.9&nbsp;mm [+/-0 ht]
* Open diapason (broader-toned principal stop): 57.4&nbsp;mm [+1 ht]
* Gedeckt (thin-toned flute stop): 65.4&nbsp;mm [+4 ht]
* Flûte à cheminée (typical mid-scale flute stop): 74.4&nbsp;mm [+7 ht]
* Flûte ouverte (broader-toned flute stop): 81.1&nbsp;mm [+9 ht]

Normalmensur scaling table, 17th halving ratio:

{| class="wikitable"
|-
! !! !!32&prime;!!!!16&prime;!!!!8&prime;!!!!4&prime;!!!!2&prime;!!!!1&prime;!!!!{{1/2}}&prime;!!!!{{1/4}}&prime;!!!!{{Frac|1|8}}&prime;!!!!{{Frac|1|16}}&prime;
|-
! !!mm!!scale!!mm!!scale!!mm!!scale!!mm!!scale!!mm!!scale!!mm!!scale!!mm!!scale!!mm!!scale!!mm!!scale!!mm!!scale

|-
|C 1||439.7||20||261.5||32||155.5||44||92.4||56||54.9||68||32.6||80||19.3||92||11.5||104||6.8||116||4.0||128
|-
|C# 2||421.2||21||250.4||33||148.9||45||88.5||57||52.6||69||31.3||81||18.6||93||11.0||105||6.5||117||3.9||129
|-
|D 3||403.2||22||239.8||34||142.6||46||84.7||58||50.4||70||29.9||82||17.8||94||10.5||106||6.3||118||3.7||130
|-
|D# 4||386.2||23||229.6||35||136.5||47||81.1||59||48.2||71||28.7||83||16.9||95||10.1||107||6.0||119||3.6||131
|-
|E 5||369.9||24||219.9||36||130.7||48||77.7||60||46.2||72||27.4||84||16.3||96||9.7||108||5.7||120||3.4||132
|-
|F 6||354.1||25||210.6||37||125.2||49||74.4||61||44.2||73||26.3||85||15.6||97||9.3||109||5.5||121||3.3||133
|-
|F# 7||339.1||26||201.6||38||119.9||50||71.3||62||42.3||74||25.2||86||14.9||98||8.8||110||5.2||122||3.1||134
|-
|G 8||324.7||27||193.1||39||114.8||51||68.2||63||40.5||75||24.1||87||14.3||99||8.5||111||5.0||123||3.0||135
|-
|G# 9||311.0||28||184.9||40||109.9||52||65.3||64||38.8||76||23.1||88||13.7||100||8.1||112||4.8||124||2.8||136
|-
|A 10||297.8||29||177.1||41||105.3||53||62.6||65||37.2||77||22.1||89||13.1||101||7.8||113||4.6||125||2.7||137
|-
|A# 11||285.2||30||169.5||42||100.8||54||59.9||66||35.6||78||21.1||90||12.6||102||7.4||114||4.4||126||2.6||138
|-
|B 12||273.1||31||162.3||43||96.5||55||57.4||67||34.1||79||20.2||91||12.0||103||7.1||115||4.2||127||2.5||139
|}

[http://www.organsupply.com/assets/CatalogPDFs/Section-9-Pipes-and-Supplies.pdf From Organ Supply Industries catalog]