Comparison of mode notation systems
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==__**Proposed method of naming all possible rank-2 scales**__== **This page is a work in progress...** Question: number MODMOS modes by position in compacted or uncompacted genchain? Question: resolve the ambiguity for MODMOS mode numbers? **Mode numbers** provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[xenharmonic/Modal UDP notation|Modal UDP notation]], it starts with the convention of using //some-temperament-name//[//some-number//] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator: || old scale name || new scale name || Ls pattern || example on white keys || genchain || || Lydian || 1st Meantone [7] || LLLs LLs || F G A B C D E F || __**F**__ C G D A E B || || Ionian (major) || 2nd Meantone [7] || LLsL LLs || C D E F G A B C || F __**C**__ G D A E B || || Mixolydian || 3rd Meantone [7] || LLsL LsL || G A B C D E F G || F C __**G**__ D A E B || || Dorian || 4th Meantone [7] || LsLL LsL || D E F G A B C D || F C G __**D**__ A E B || || Aeolian (minor) || 5th Meantone [7] || LsLL sLL || A B C D E F G A || F C G D __**A**__ E B || || Phrygian || 6th Meantone [7] || sLLL sLL || E F G A B C D E || F C G D A __**E**__ B || || Locrian || 7th Meantone [7] || sLLs LLL || B C D E F G A B || F C G D A E __**B**__ || These [[MOSScales|MOS scales]] are formed from a segment of the [[periods and generators|generator-chain]], or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions. 4th Meantone [7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic". The same seven modes, all with C as the tonic, to illustrate the difference between modes. Adjacent modes differ by only one note. The modes proceed from sharper (Lydian) to flatter (Locrian). || old scale name || new scale name || Ls pattern || example in C || ------------------- genchain --------------- || || Lydian || 1st Meantone [7] || LLLs LLs || C D E F# G A B C ||> __**C**__ G D A E B F# || || Ionian (major) || 2nd Meantone [7] || LLsL LLs || C D E F G A B C ||> F __**C**__ G D A E B ---- || || Mixolydian || 3rd Meantone [7] || LLsL LsL || C D E F G A Bb C ||> Bb F __**C**__ G D A E ------- || || Dorian || 4th Meantone [7] || LsLL LsL || C D Eb F G A Bb C || ------------- Eb Bb F __**C**__ G D A || || Aeolian (minor) || 5th Meantone [7] || LsLL sLL || C D Eb F G Ab Bb C || --------- Ab Eb Bb F __**C**__ G D || || Phrygian || 6th Meantone [7] || sLLL sLL || C Db Eb F G Ab Bb C || ---- Db Ab Eb Bb F __**C**__ G || || Locrian || 7th Meantone [7] || sLLs LLL || C Db Eb F Gb Ab Bb C || Gb Db Ab Eb Bb F __**C**__ || The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. **Unlike modal UDP notation, the generator isn't always chroma-positive.** This is necessary to keep the same generator for different MOS's of the same [[Regular Temperaments|temperament]], which guarantees that the smaller MOS will always be a subset of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes proceed from flatter to sharper. Because Meantone [5] and Meantone [7]have the same generator, C 2nd Meantone [5] = CDFGAC is a subset of C 2nd Meantone [7] = CDEFGABC. Pentatonic meantone scales: || old scale name || new scale name || Ls pattern || example in C || --------- genchain ------- || || major pentatonic || 1st Meantone [5] || ssL sL || C D E G A C ||> __**C**__ G D A E || ||= || 2nd Meantone [5] || sLs sL || C D F G A C ||> F __**C**__ G D A -- || ||= || 3rd Meantone [5] || sLs Ls || C D F G Bb C || -------- Bb F __**C**__ G D || || minor pentatonic || 4th Meantone [5] || Lss Ls || C Eb F G Bb C || ---- Eb Bb F __**C**__ G || ||= || 5th Meantone [5] || LsL ss || C Eb F Ab Bb C || Ab Eb Bb F __**C**__ || Chromatic meantone scales. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged. || scale name || Ls pattern || example in C || genchain || || 1st Meantone [12] || sLsLsLL sLsLL || C C# D D# E E# F# G G# A A# B C || __**C**__ G D A E B F# C# G# D# A# E# || || 2nd Meantone [12] || sLsLLsL sLsLL || C C# D D# E F F# G G# A A# B C || F __**C**__ G D A E B F# C# G# D# A# || || 3rd Meantone [12] || sLsLLsL sLLsL || C C# D D# E F F# G G# A Bb B C || Bb F __**C**__ G D A E B F# C# G# D# || || 4th Meantone [12] || sLLsLsL sLLsL || C C# D Eb E F F# G G# A Bb B C || Eb Bb F __**C**__ G D A E B F# C# G# || || 5th Meantone [12] || sLLsLsL LsLsL || C C# D Eb E F F# G Ab A Bb B C || Ab Eb Bb F __**C**__ G D A E B F# C# || || 6th Meantone [12] || LsLsLsL LsLsL || C Db D Eb E F F# G Ab A Bb B C || Db Ab Eb Bb F __**C**__ G D A E B F# || || 7th Meantone [12] || LsLsLLs LsLsL || C Db D Eb E F Gb G Ab A Bb B C || Gb Db Ab Eb Bb F __**C**__ G D A E B || ||= etc. || || || || [[Sensi]] [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19) || scale name || Ls pattern || example in C || genchain || || 1st Sensi [8] || ssL ssL sL || C Db D# E# F# G A Bb C || __**C**__ E# A Db F# Bb D# G || || 2nd Sensi [8] || ssL sL ssL || C Db D# E# F# G# A Bb C || G# __**C**__ E# A Db F# Bb D# || || 3rd Sensi [8] || sL ssL ssL || C Db Eb E# F# G# A Bb C || Eb G# __**C**__ E# A Db F# Bb || || 4th Sensi [8] || sL ssL sL s || C Db Eb E# F# G# A B C || B Eb G# __**C**__ E# A Db F# || || 5th Sensi [8] || sL sL ssL s || C Db Eb E# Gb G# A B C || Gb B Eb G# __**C**__ E# A Db || || 6th Sensi [8] || Lss Lss Ls || C D Eb E# Gb G# A B C || D Gb B Eb G# __**C**__ E# A || || 7th Sensi [8] || Lss Ls Lss || C D Eb E# Gb G# A# B C || A# D Gb B Eb G# __**C**__ E# || || 8th Sensi [8] || Ls Lss Lss || C D Eb F Gb G# A# B C || F A# D Gb B Eb G# __**C**__ || Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[xenharmonic/ups and downs notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale. || scale name || Ls pattern || example in C || genchain || || 1st Porcupine [7] || ssss ssL || C Dv Eb^ F Gv Ab^ Bb C || __**C**__ Dv Eb^ F Gv Ab^ Bb || || 2nd Porcupine [7] || ssss sLs || C Dv Eb^ F Gv Ab^ Bb^ C || Bb^ __**C**__ Dv Eb^ F Gv Ab^ || || 3rd Porcupine [7] || ssss Lss || C Dv Eb^ F Gv Av Bb^ C || Av Bb^ __**C**__ Dv Eb^ F Gv || || 4th Porcupine [7] || sssL sss || C Dv Eb^ F G Av Bb^ C || G Av Bb^ __**C**__ Dv Eb^ F || || 5th Porcupine [7] || ssLs sss || C Dv Eb^ F^ G Av Bb^ C ||= F^ G Av Bb^ __**C**__ Dv Eb^ || || 6th Porcupine [7] || sLss sss || C Dv Ev F^ G Av Bb^ C || Ev F^ G Av Bb^ __**C**__ Dv || || 7th Porcupine [7] || Lsss sss || C D Ev F^ G Av Bb^ C || D Ev F^ G Av Bb^ __**C**__ || ==[[#How to name rank-2 scales-MODMOS scales]]**__MODMOS scales__**== To find a [[MODMOS Scales|MODMOS]] scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the __compacted__ genchain. //[This may change]// For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. This is so that two notes an aug or dim fifth apart will have adjacent mode numbers. Just like A and E are adjacent, Ab and E are too. In other words, determining the mode number from the scale degree remains fifth-based. Meantone [7,+3,-6] means that the 3rd note in the __compacted__ genchain is moved 7 steps to the right, and the 6th note is moved 7 steps to the left. The alterations are the exact opposite of the alterations needed to close the gaps in the uncompacted genchain. "+" and "-" are preferred over "#" and "b" because in the case of a chroma-negative generator, "+" makes the note flatter, as in the last example: || old scale name || example in A || genchain || compacted genchain || new scale name || || Harmonic minor || A B C D E F G# A || F C * D __**A**__ E B * * G# || F C G D __**A**__ E B || 5th Meantone [7,+3] || || Ascending melodic minor || A B C D E F# G# A || C * D __**A**__ E B F# * G# || F C G D __**A**__ E B || 5th Meantone [7,+1,+3] || ||= " ||= " ||= " || C G D __**A**__ E B F# || 4th Meantone [7,+2] || ||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone [7,-6] || || Double harmonic minor || A B C D# E F G# A || F C * * __**A**__ E B * * G# D# || F C G D __**A**__ E B || 5th Meantone [7,+3,+4] || ||= " ||= " ||= " || __**A**__ E B F# C# G# D# || 1st Meantone [7,-4,-5] || || Double harmonic major || A Bb C# D E F G# A || Bb F * * D __**A**__ E * * C# G# || Bb F C G D __**A**__ E || 6th Meantone [7,+3,+4] || ||= " ||= " ||= " || D __**A**__ E B F# C# G# || 2nd Meantone [7,-4,-5] || || <span class="mw-redirect">Hungarian gypsy </span>minor || A B C D# E F G A || F C G * __**A**__ E B * * * D# || F C G D __**A**__ E B || 5th Meantone [7,+4] || || Phrygian dominant || A Bb C# D E F G A || Bb F * G D __**A**__ E * * C# || Bb F C G D __**A**__ E || 6th Meantone [7,+3] || || a pentatonic scale || C D E G A# || A# * __**C**__ G D * E || __**C**__ G D A E || 1st Meantone [5,-4] || ||= " ||= " ||= " || A# E# __**C**__ G D || 3rd Meantone [5,+2] || The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right. The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is called 1st Meantone [7,-4,-5] and the double harmonic major is 6th Meantone [7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone [7] to melodic minor = 5th Meantone [7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale. Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes: X X * X X X X * * X 1: C D# E F# G A B C 2: C D E F G# A B C 3: C Db Eb Fb Gb Ab Bbb C 4: C D Eb F# G A Bb C 5: C D Eb F G Ab B C 6: C Db E F G Ab Bb C 7: C Db Eb F Gb A Bb C Melodic minor modes: 1: C D E F# G# A B C 2: C Db Eb Fb Gb Ab Bb C 3: C D E F# G A Bb C 4: C D Eb F G A B C 5: C D E F G Ab Bb C 6: C Db Eb F G A Bb C 7: C D Eb F Gb Ab Bb C ==[[#How to name rank-2 scales-Fractional-octave periods]]**__Fractional-octave periods__**== Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this: Eb -- Bb -- F --- C --- G A --- E --- B --- F# -- C# Or alternatively, using 16/15 not 3/2 as the generator: Eb -- E --- F --- F# -- G A --- Bb -- B --- C --- C# Multiple genchains occur because rank-2 really is 2 dimensional, with a "genweb" running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. When the period is an octave, this octave-reduces to a single horizontal genchain. But shrutal has a genweb with vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete. If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either 3/2 or 16/15, because 16/15 would still create the same mode numbers and thus the same scale names. All five Shrutal [10] modes: || scale name || Ls pattern || example in C || 1st genchain || 2nd genchain || || 1st Shrutal [10] || ssssL-ssssL || C C# D D# E F# G G# A A# C || __**C**__ G D A E || F# C# G# D# A# || || 2nd Shrutal [10] || sssLs-sssLs || C C# D D# F F# G G# A B C || F __**C**__ G D A || B F# C# G# D# || || 3rd Shrutal [10] || ssLss-ssLss || C C# D E F F# G G# Bb B C || Bb F __**C**__ G D || E B F# C# G# || || 4th Shrutal [10] || sLsss-sLsss || C C# Eb E F F# G A Bb B C || Eb Bb F __**C**__ G || A E B F# C# || || 5th Shrutal [10] || Lssss-Lssss || C D Eb E F F# Ab A Bb B C || Ab Eb Bb F __**C**__ || D A E B F# || The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains. || scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || || 1st Diminished[8] || sLsL sLsL || C Db Eb E Gb G A Bb C ||= __**C**__ G || Eb Bb || Gb Db || A E || || 2nd Diminished[8] || LsLs LsLs || C D Eb F F# Ab A B C ||= F __**C**__ || Ab Eb || B F# || D A || There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names. || scale name || Ls pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain || 5th chain || || 1st Blackwood[10] || LsLsLs LsLs || C C#v D Ev F F#v G Av A Bv C ||= __**C**__ Ev || D F#v || F Av || G Bv || A C#v || || 2nd Blackwood[10] || sLsLsL sLsL || C C^ D Eb^ E F^ G Ab^ A Bb^ C ||= Ab^ __**C**__ || Bb^ D || C^ E || Eb^ G || F^ A || ==[[#How to name rank-2 scales-Non-MOS scales]]**__Non-MOS non-MODMOS scales__**== Compact the genchain to remove any gaps via alterations. The mode number is derived from the compacted genchain. Examples: C D E F F# G A B C has the genchain F __**C**__ G D A E B F#, and is named C 2nd Meantone[8]. C D E F F# G A Bb C, with genchain Bb F __**C**__ G D A E * F#. Alter Bb to get an unbroken genchain: F __**C**__ G D A E B F#. The scale is C 2nd Meantone[8,-7]. A B C D D# E F G G# A, with genchain F C G D __**A**__ E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D __**A**__ E B F# C# G# D#, giving the name A 3rd Meantone[9,-6,-7]. F G A C E F, with genchain __**F**__ C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6,no4]. A B C E F A, a japanese pentatonic scale, with genchain F C * * __**A**__ E B. The F and C can't be sharpened to F# and C# to make an unbroken Meantone[5] genchain, because pentatonic alterations move notes by 5 steps, not 7. No amount of __pentatonic__ altering will make an unbroken genchain, so the scale must be named as a heptatonic scale with missing degrees: A 5th Meantone[7,no3,no4]. //[Problem: The 2nd and 3rd examples use heptatonic alterations even though the scales aren't heptatonic.]// //[Problem: octotonic alterations would be absurd!]// //[Possible solution: non-MOS scales can't have chromatic alterations, only missing notes. 2nd example is 3rd Meantone[9,no8]. 3rd example is 5th Meantone[11,no8,no9].// ==[[#How to name rank-2 scales-Non-MOS scales]]__Explanation / Rationale__== ===**__Why not number the modes in the order they occur in the scale?__**=== Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc. __Genchain-based__: if the Meantone[7] genchain were notated 1 2 3 4 5 6 7, the Lydian scale would be 1 3 5 7 2 4 6 1, and the major scale would be 2 4 6 1 3 5 7 2. __Scale-based__: if the Meantone[7] major scale were notated 1 2 3 4 5 6 7 1, the genchain would be 4 1 5 2 6 3 7. The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The modes are ordered in a spectrum, and the 1st and last modes are always the two most extreme. For MOS scales, adjacent modes differ by only one note. The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain. ===__**Why make an exception for 3/2 vs 4/3 as the generator?**__=== Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show: "Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio __**<span class="mw-redirect">3:2</span>**__ (i.e., the untempered perfect __**fifth**__)." -- [[https://en.wikipedia.org/wiki/Pythagorean_tuning|en.wikipedia.org/wiki/Pythagorean_tuning]] "The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect __**fifth**__." -- [[https://en.wikipedia.org/wiki/Syntonic_temperament|en.wikipedia.org/wiki/Syntonic_temperament]] "Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect __**fifths**__." -- [[https://en.wikipedia.org/wiki/Meantone_temperament|en.wikipedia.org/wiki/Meantone_temperament]] "In this system the perfect __**fifth**__ is flattened by one quarter of a syntonic comma." -- [[https://en.wikipedia.org/wiki/Quarter-comma_meantone|en.wikipedia.org/wiki/Quarter-comma_meantone]] "The term "well temperament" or "good temperament" usually means some sort of <span class="new">irregular temperament</span> in which the tempered __**fifths**__ are of different sizes." -- [[https://en.wikipedia.org/wiki/Well_temperament|en.wikipedia.org/wiki/Well_temperament]] "A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike __wise__ consistencies, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2. ===__**Then why not always choose the larger of the two generators?**__=== Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.) ===__Why not always choose the chroma-positive generator?__=== See below. ===__**Why not just use UDP notation?**__=== One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. || scale || UDP generator || UDP genchain || Mode Numbers generator || Mode Numbers genchain || || Meantone[5] in 31edo ||= 4/3 || E A D G C ||= 3/2 || C G D A E || || Meantone[7] in 31edo ||= 3/2 || C G D A E B F# ||= 3/2 || C G D A E B F# || || Meantone[12] in 31edo ||= 4/3 || E# A# D# G# C# F# B E A D G C ||= 3/2 || C G D A E B F# C# G# D# A# E# || || Meantone[19] in 31edo ||= 3/2 || C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex ||= 3/2 || C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex || A larger problem is that choosing the chroma-positive generator only applies to MOS and MODMOS scales, and breaks down when the length of the genchain results in a non-MOS scale. Mode Numbers notation can be applied to scales like Meantone[8], which while not a MOS, is certainly musically useful. || scale || UDP genchain || Mode Numbers genchain || || Meantone [2] || C G || C G || || Meantone [3] || D G C || C G D || || Meantone [4] || ??? || C G D A || || Meantone [5] || E A D G C || C G D A E || || Meantone [6] || ??? || G C D A E B || || Meantone [7] || C G D A E B F# || C G D A E B F# || || Meantone [8] || ??? || C G D A E B F# C# || || Meantone [9] || ??? || C G D A E B F# C# G# || || Meantone [10] || ??? || C G D A E B F# C# G# D# || || Meantone [11] || ??? || C G D A E B F# C# G# D# A# || || Meantone [12] if generator < 700¢ || E# A# D# G# C# F# B E A D G C || C G D A E B F# C# G# D# A# E# || ||< Meantone [12] if generator > 700¢ || C G D A E B F# C# G# D# A# E# ||= C G D A E B F# C# G# D# A# E# || An even larger problem is that the notation is overly tuning-dependent. Meantone[12] generated by 701¢ has a different genchain than Meantone[12] generated by 699¢, so slight differences in tempering result in different mode names. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth[12]'s fifth to be more than 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either more or less than 700¢. This makes every single UDP mode of Dominant[12] ambiguous. For example "Dominant 8|3" could mean either "4th Dominant[12]" or "9th Dominant[12]". Something similar happens with Meantone[19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone[19] ambiguous. Another example is Dicot[7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore[5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it. A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical [[https://en.wikipedia.org/wiki/Zero-based_numbering|zero-based counting]] and Mode Numbers notation uses the more intuitive one-based counting. Admins: Please delete the blank page at http://xenharmonic.wikispaces.com/Modal+Notation+by+Genchain+Position
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<strong>This page is a work in progress...</strong><br />
Question: number MODMOS modes by position in compacted or uncompacted genchain?<br />
Question: resolve the ambiguity for MODMOS mode numbers?<br />
<br />
<strong>Mode numbers</strong> provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Modal%20UDP%20notation">Modal UDP notation</a>, it starts with the convention of using <em>some-temperament-name</em>[<em>some-number</em>] to create a generator-chain, and adds a way to number each mode uniquely. For example, here are all the modes of Meantone[7], using ~3/2 as the generator:<br />
<table class="wiki_table">
<tr>
<td>old scale name<br />
</td>
<td>new scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example on white keys<br />
</td>
<td>genchain<br />
</td>
</tr>
<tr>
<td>Lydian<br />
</td>
<td>1st Meantone [7]<br />
</td>
<td>LLLs LLs<br />
</td>
<td>F G A B C D E F<br />
</td>
<td><u><strong>F</strong></u> C G D A E B<br />
</td>
</tr>
<tr>
<td>Ionian (major)<br />
</td>
<td>2nd Meantone [7]<br />
</td>
<td>LLsL LLs<br />
</td>
<td>C D E F G A B C<br />
</td>
<td>F <u><strong>C</strong></u> G D A E B<br />
</td>
</tr>
<tr>
<td>Mixolydian<br />
</td>
<td>3rd Meantone [7]<br />
</td>
<td>LLsL LsL<br />
</td>
<td>G A B C D E F G<br />
</td>
<td>F C <u><strong>G</strong></u> D A E B<br />
</td>
</tr>
<tr>
<td>Dorian<br />
</td>
<td>4th Meantone [7]<br />
</td>
<td>LsLL LsL<br />
</td>
<td>D E F G A B C D<br />
</td>
<td>F C G <u><strong>D</strong></u> A E B<br />
</td>
</tr>
<tr>
<td>Aeolian (minor)<br />
</td>
<td>5th Meantone [7]<br />
</td>
<td>LsLL sLL<br />
</td>
<td>A B C D E F G A<br />
</td>
<td>F C G D <u><strong>A</strong></u> E B<br />
</td>
</tr>
<tr>
<td>Phrygian<br />
</td>
<td>6th Meantone [7]<br />
</td>
<td>sLLL sLL<br />
</td>
<td>E F G A B C D E<br />
</td>
<td>F C G D A <u><strong>E</strong></u> B<br />
</td>
</tr>
<tr>
<td>Locrian<br />
</td>
<td>7th Meantone [7]<br />
</td>
<td>sLLs LLL<br />
</td>
<td>B C D E F G A B<br />
</td>
<td>F C G D A E <u><strong>B</strong></u><br />
</td>
</tr>
</table>
<br />
These <a class="wiki_link" href="/MOSScales">MOS scales</a> are formed from a segment of the <a class="wiki_link" href="/periods%20and%20generators">generator-chain</a>, or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions. 4th Meantone [7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic".<br />
<br />
The same seven modes, all with C as the tonic, to illustrate the difference between modes. Adjacent modes differ by only one note. The modes proceed from sharper (Lydian) to flatter (Locrian).<br />
<table class="wiki_table">
<tr>
<td>old scale name<br />
</td>
<td>new scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>------------------- genchain ---------------<br />
</td>
</tr>
<tr>
<td>Lydian<br />
</td>
<td>1st Meantone [7]<br />
</td>
<td>LLLs LLs<br />
</td>
<td>C D E F# G A B C<br />
</td>
<td style="text-align: right;"><u><strong>C</strong></u> G D A E B F#<br />
</td>
</tr>
<tr>
<td>Ionian (major)<br />
</td>
<td>2nd Meantone [7]<br />
</td>
<td>LLsL LLs<br />
</td>
<td>C D E F G A B C<br />
</td>
<td style="text-align: right;">F <u><strong>C</strong></u> G D A E B ----<br />
</td>
</tr>
<tr>
<td>Mixolydian<br />
</td>
<td>3rd Meantone [7]<br />
</td>
<td>LLsL LsL<br />
</td>
<td>C D E F G A Bb C<br />
</td>
<td style="text-align: right;">Bb F <u><strong>C</strong></u> G D A E -------<br />
</td>
</tr>
<tr>
<td>Dorian<br />
</td>
<td>4th Meantone [7]<br />
</td>
<td>LsLL LsL<br />
</td>
<td>C D Eb F G A Bb C<br />
</td>
<td>------------- Eb Bb F <u><strong>C</strong></u> G D A<br />
</td>
</tr>
<tr>
<td>Aeolian (minor)<br />
</td>
<td>5th Meantone [7]<br />
</td>
<td>LsLL sLL<br />
</td>
<td>C D Eb F G Ab Bb C<br />
</td>
<td>--------- Ab Eb Bb F <u><strong>C</strong></u> G D<br />
</td>
</tr>
<tr>
<td>Phrygian<br />
</td>
<td>6th Meantone [7]<br />
</td>
<td>sLLL sLL<br />
</td>
<td>C Db Eb F G Ab Bb C<br />
</td>
<td>---- Db Ab Eb Bb F <u><strong>C</strong></u> G<br />
</td>
</tr>
<tr>
<td>Locrian<br />
</td>
<td>7th Meantone [7]<br />
</td>
<td>sLLs LLL<br />
</td>
<td>C Db Eb F Gb Ab Bb C<br />
</td>
<td>Gb Db Ab Eb Bb F <u><strong>C</strong></u><br />
</td>
</tr>
</table>
<br />
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons. <strong>Unlike modal UDP notation, the generator isn't always chroma-positive.</strong> This is necessary to keep the same generator for different MOS's of the same <a class="wiki_link" href="/Regular%20Temperaments">temperament</a>, which guarantees that the smaller MOS will always be a subset of the larger MOS.<br />
<br />
For example, Meantone [5] is generated by 3/2, not 4/3. Because the generator is chroma-negative, the modes proceed from flatter to sharper. Because Meantone [5] and Meantone [7]have the same generator, C 2nd Meantone [5] = CDFGAC is a subset of C 2nd Meantone [7] = CDEFGABC.<br />
<br />
Pentatonic meantone scales:<br />
<table class="wiki_table">
<tr>
<td>old scale name<br />
</td>
<td>new scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>--------- genchain -------<br />
</td>
</tr>
<tr>
<td>major pentatonic<br />
</td>
<td>1st Meantone [5]<br />
</td>
<td>ssL sL<br />
</td>
<td>C D E G A C<br />
</td>
<td style="text-align: right;"><u><strong>C</strong></u> G D A E<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td>2nd Meantone [5]<br />
</td>
<td>sLs sL<br />
</td>
<td>C D F G A C<br />
</td>
<td style="text-align: right;">F <u><strong>C</strong></u> G D A --<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td>3rd Meantone [5]<br />
</td>
<td>sLs Ls<br />
</td>
<td>C D F G Bb C<br />
</td>
<td>-------- Bb F <u><strong>C</strong></u> G D<br />
</td>
</tr>
<tr>
<td>minor pentatonic<br />
</td>
<td>4th Meantone [5]<br />
</td>
<td>Lss Ls<br />
</td>
<td>C Eb F G Bb C<br />
</td>
<td>---- Eb Bb F <u><strong>C</strong></u> G<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><br />
</td>
<td>5th Meantone [5]<br />
</td>
<td>LsL ss<br />
</td>
<td>C Eb F Ab Bb C<br />
</td>
<td>Ab Eb Bb F <u><strong>C</strong></u><br />
</td>
</tr>
</table>
<br />
Chromatic meantone scales. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.<br />
<table class="wiki_table">
<tr>
<td>scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>genchain<br />
</td>
</tr>
<tr>
<td>1st Meantone [12]<br />
</td>
<td>sLsLsLL sLsLL<br />
</td>
<td>C C# D D# E E# F# G G# A A# B C<br />
</td>
<td><u><strong>C</strong></u> G D A E B F# C# G# D# A# E#<br />
</td>
</tr>
<tr>
<td>2nd Meantone [12]<br />
</td>
<td>sLsLLsL sLsLL<br />
</td>
<td>C C# D D# E F F# G G# A A# B C<br />
</td>
<td>F <u><strong>C</strong></u> G D A E B F# C# G# D# A#<br />
</td>
</tr>
<tr>
<td>3rd Meantone [12]<br />
</td>
<td>sLsLLsL sLLsL<br />
</td>
<td>C C# D D# E F F# G G# A Bb B C<br />
</td>
<td>Bb F <u><strong>C</strong></u> G D A E B F# C# G# D#<br />
</td>
</tr>
<tr>
<td>4th Meantone [12]<br />
</td>
<td>sLLsLsL sLLsL<br />
</td>
<td>C C# D Eb E F F# G G# A Bb B C<br />
</td>
<td>Eb Bb F <u><strong>C</strong></u> G D A E B F# C# G#<br />
</td>
</tr>
<tr>
<td>5th Meantone [12]<br />
</td>
<td>sLLsLsL LsLsL<br />
</td>
<td>C C# D Eb E F F# G Ab A Bb B C<br />
</td>
<td>Ab Eb Bb F <u><strong>C</strong></u> G D A E B F# C#<br />
</td>
</tr>
<tr>
<td>6th Meantone [12]<br />
</td>
<td>LsLsLsL LsLsL<br />
</td>
<td>C Db D Eb E F F# G Ab A Bb B C<br />
</td>
<td>Db Ab Eb Bb F <u><strong>C</strong></u> G D A E B F#<br />
</td>
</tr>
<tr>
<td>7th Meantone [12]<br />
</td>
<td>LsLsLLs LsLsL<br />
</td>
<td>C Db D Eb E F Gb G Ab A Bb B C<br />
</td>
<td>Gb Db Ab Eb Bb F <u><strong>C</strong></u> G D A E B<br />
</td>
</tr>
<tr>
<td style="text-align: center;">etc.<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<a class="wiki_link" href="/Sensi">Sensi</a> [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)<br />
<table class="wiki_table">
<tr>
<td>scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>genchain<br />
</td>
</tr>
<tr>
<td>1st Sensi [8]<br />
</td>
<td>ssL ssL sL<br />
</td>
<td>C Db D# E# F# G A Bb C<br />
</td>
<td><u><strong>C</strong></u> E# A Db F# Bb D# G<br />
</td>
</tr>
<tr>
<td>2nd Sensi [8]<br />
</td>
<td>ssL sL ssL<br />
</td>
<td>C Db D# E# F# G# A Bb C<br />
</td>
<td>G# <u><strong>C</strong></u> E# A Db F# Bb D#<br />
</td>
</tr>
<tr>
<td>3rd Sensi [8]<br />
</td>
<td>sL ssL ssL<br />
</td>
<td>C Db Eb E# F# G# A Bb C<br />
</td>
<td>Eb G# <u><strong>C</strong></u> E# A Db F# Bb<br />
</td>
</tr>
<tr>
<td>4th Sensi [8]<br />
</td>
<td>sL ssL sL s<br />
</td>
<td>C Db Eb E# F# G# A B C<br />
</td>
<td>B Eb G# <u><strong>C</strong></u> E# A Db F#<br />
</td>
</tr>
<tr>
<td>5th Sensi [8]<br />
</td>
<td>sL sL ssL s<br />
</td>
<td>C Db Eb E# Gb G# A B C<br />
</td>
<td>Gb B Eb G# <u><strong>C</strong></u> E# A Db<br />
</td>
</tr>
<tr>
<td>6th Sensi [8]<br />
</td>
<td>Lss Lss Ls<br />
</td>
<td>C D Eb E# Gb G# A B C<br />
</td>
<td>D Gb B Eb G# <u><strong>C</strong></u> E# A<br />
</td>
</tr>
<tr>
<td>7th Sensi [8]<br />
</td>
<td>Lss Ls Lss<br />
</td>
<td>C D Eb E# Gb G# A# B C<br />
</td>
<td>A# D Gb B Eb G# <u><strong>C</strong></u> E#<br />
</td>
</tr>
<tr>
<td>8th Sensi [8]<br />
</td>
<td>Ls Lss Lss<br />
</td>
<td>C D Eb F Gb G# A# B C<br />
</td>
<td>F A# D Gb B Eb G# <u><strong>C</strong></u><br />
</td>
</tr>
</table>
<br />
Porcupine [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using <a class="wiki_link" href="http://xenharmonic.wikispaces.com/ups%20and%20downs%20notation">ups and downs notation</a>.<br />
Because the generator is a 2nd, the genchain resembles the scale.<br />
<table class="wiki_table">
<tr>
<td>scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>genchain<br />
</td>
</tr>
<tr>
<td>1st Porcupine [7]<br />
</td>
<td>ssss ssL<br />
</td>
<td>C Dv Eb^ F Gv Ab^ Bb C<br />
</td>
<td><u><strong>C</strong></u> Dv Eb^ F Gv Ab^ Bb<br />
</td>
</tr>
<tr>
<td>2nd Porcupine [7]<br />
</td>
<td>ssss sLs<br />
</td>
<td>C Dv Eb^ F Gv Ab^ Bb^ C<br />
</td>
<td>Bb^ <u><strong>C</strong></u> Dv Eb^ F Gv Ab^<br />
</td>
</tr>
<tr>
<td>3rd Porcupine [7]<br />
</td>
<td>ssss Lss<br />
</td>
<td>C Dv Eb^ F Gv Av Bb^ C<br />
</td>
<td>Av Bb^ <u><strong>C</strong></u> Dv Eb^ F Gv<br />
</td>
</tr>
<tr>
<td>4th Porcupine [7]<br />
</td>
<td>sssL sss<br />
</td>
<td>C Dv Eb^ F G Av Bb^ C<br />
</td>
<td>G Av Bb^ <u><strong>C</strong></u> Dv Eb^ F<br />
</td>
</tr>
<tr>
<td>5th Porcupine [7]<br />
</td>
<td>ssLs sss<br />
</td>
<td>C Dv Eb^ F^ G Av Bb^ C<br />
</td>
<td style="text-align: center;">F^ G Av Bb^ <u><strong>C</strong></u> Dv Eb^<br />
</td>
</tr>
<tr>
<td>6th Porcupine [7]<br />
</td>
<td>sLss sss<br />
</td>
<td>C Dv Ev F^ G Av Bb^ C<br />
</td>
<td>Ev F^ G Av Bb^ <u><strong>C</strong></u> Dv<br />
</td>
</tr>
<tr>
<td>7th Porcupine [7]<br />
</td>
<td>Lsss sss<br />
</td>
<td>C D Ev F^ G Av Bb^ C<br />
</td>
<td>D Ev F^ G Av Bb^ <u><strong>C</strong></u><br />
</td>
</tr>
</table>
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:20:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@How to name rank-2 scales-MODMOS scales" title="Anchor: How to name rank-2 scales-MODMOS scales"/> --><a name="How to name rank-2 scales-MODMOS scales"></a><!-- ws:end:WikiTextAnchorRule:20 --><strong><u>MODMOS scales</u></strong></h2>
To find a <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scale's name, start with the genchain for the scale, which will always have gaps. Compact it into a chain without gaps by altering one or more notes. If there is more than one way to do this, the way that alters as few notes as possible is generally preferable. Determine the mode number from the <u>compacted</u> genchain. <em>[This may change]</em> For example, for harmonic minor, A is the 4th note of the uncompacted genchain, but the 5th note of the compacted one. This is so that two notes an aug or dim fifth apart will have adjacent mode numbers. Just like A and E are adjacent, Ab and E are too. In other words, determining the mode number from the scale degree remains fifth-based.<br />
<br />
Meantone [7,+3,-6] means that the 3rd note in the <u>compacted</u> genchain is moved 7 steps to the right, and the 6th note is moved 7 steps to the left. The alterations are the exact opposite of the alterations needed to close the gaps in the uncompacted genchain. "+" and "-" are preferred over "#" and "b" because in the case of a chroma-negative generator, "+" makes the note flatter, as in the last example:<br />
<table class="wiki_table">
<tr>
<td>old scale name<br />
</td>
<td>example in A<br />
</td>
<td>genchain<br />
</td>
<td>compacted genchain<br />
</td>
<td>new scale name<br />
</td>
</tr>
<tr>
<td>Harmonic minor<br />
</td>
<td>A B C D E F G# A<br />
</td>
<td>F C * D <u><strong>A</strong></u> E B * * G#<br />
</td>
<td>F C G D <u><strong>A</strong></u> E B<br />
</td>
<td>5th Meantone [7,+3]<br />
</td>
</tr>
<tr>
<td>Ascending melodic minor<br />
</td>
<td>A B C D E F# G# A<br />
</td>
<td>C * D <u><strong>A</strong></u> E B F# * G#<br />
</td>
<td>F C G D <u><strong>A</strong></u> E B<br />
</td>
<td>5th Meantone [7,+1,+3]<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td>C G D <u><strong>A</strong></u> E B F#<br />
</td>
<td>4th Meantone [7,+2]<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td>D <u><strong>A</strong></u> E B F# C# G#<br />
</td>
<td>2nd Meantone [7,-6]<br />
</td>
</tr>
<tr>
<td>Double harmonic minor<br />
</td>
<td>A B C D# E F G# A<br />
</td>
<td>F C * * <u><strong>A</strong></u> E B * * G# D#<br />
</td>
<td>F C G D <u><strong>A</strong></u> E B<br />
</td>
<td>5th Meantone [7,+3,+4]<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td><u><strong>A</strong></u> E B F# C# G# D#<br />
</td>
<td>1st Meantone [7,-4,-5]<br />
</td>
</tr>
<tr>
<td>Double harmonic major<br />
</td>
<td>A Bb C# D E F G# A<br />
</td>
<td>Bb F * * D <u><strong>A</strong></u> E * * C# G#<br />
</td>
<td>Bb F C G D <u><strong>A</strong></u> E<br />
</td>
<td>6th Meantone [7,+3,+4]<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td>D <u><strong>A</strong></u> E B F# C# G#<br />
</td>
<td>2nd Meantone [7,-4,-5]<br />
</td>
</tr>
<tr>
<td><span class="mw-redirect">Hungarian gypsy </span>minor<br />
</td>
<td>A B C D# E F G A<br />
</td>
<td>F C G * <u><strong>A</strong></u> E B * * * D#<br />
</td>
<td>F C G D <u><strong>A</strong></u> E B<br />
</td>
<td>5th Meantone [7,+4]<br />
</td>
</tr>
<tr>
<td>Phrygian dominant<br />
</td>
<td>A Bb C# D E F G A<br />
</td>
<td>Bb F * G D <u><strong>A</strong></u> E * * C#<br />
</td>
<td>Bb F C G D <u><strong>A</strong></u> E<br />
</td>
<td>6th Meantone [7,+3]<br />
</td>
</tr>
<tr>
<td>a pentatonic scale<br />
</td>
<td>C D E G A#<br />
</td>
<td>A# * <u><strong>C</strong></u> G D * E<br />
</td>
<td><u><strong>C</strong></u> G D A E<br />
</td>
<td>1st Meantone [5,-4]<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td>A# E# <u><strong>C</strong></u> G D<br />
</td>
<td>3rd Meantone [5,+2]<br />
</td>
</tr>
</table>
<br />
The ambiguity of MODMOS names can be resolved by devising a rule to determine the one proper compacted genchain. For example, choose the one that moves as few notes as possible, breaking ties with a bias towards moving to the right.<br />
<br />
The disadvantage of ambiguity is that it makes modes less apparent. If the double harmonic minor is called 1st Meantone [7,-4,-5] and the double harmonic major is 6th Meantone [7,+3,+4], one can't tell that they are modes of each other. The advantage is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from minor = 5th Meantone [7] to melodic minor = 5th Meantone [7,+1,+3]. In this context, melodic minor is better described as an altered minor scale than an altered dorian scale.<br />
<br />
Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:<br />
X X * X X X X * * X<br />
1: C D# E F# G A B C<br />
2: C D E F G# A B C<br />
3: C Db Eb Fb Gb Ab Bbb C<br />
4: C D Eb F# G A Bb C<br />
5: C D Eb F G Ab B C<br />
6: C Db E F G Ab Bb C<br />
7: C Db Eb F Gb A Bb C<br />
<br />
Melodic minor modes:<br />
1: C D E F# G# A B C<br />
2: C Db Eb Fb Gb Ab Bb C<br />
3: C D E F# G A Bb C<br />
4: C D Eb F G A B C<br />
5: C D E F G Ab Bb C<br />
6: C Db Eb F G A Bb C<br />
7: C D Eb F Gb Ab Bb C<br />
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Fractional-octave periods"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:21:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@How to name rank-2 scales-Fractional-octave periods" title="Anchor: How to name rank-2 scales-Fractional-octave periods"/> --><a name="How to name rank-2 scales-Fractional-octave periods"></a><!-- ws:end:WikiTextAnchorRule:21 --><strong><u>Fractional-octave periods</u></strong></h2>
Fractional-period rank-2 temperaments have multiple genchains running in parallel. For example, shrutal[10] might look like this:<br />
Eb -- Bb -- F --- C --- G<br />
A --- E --- B --- F# -- C#<br />
<br />
Or alternatively, using 16/15 not 3/2 as the generator:<br />
Eb -- E --- F --- F# -- G<br />
A --- Bb -- B --- C --- C#<br />
<br />
Multiple genchains occur because rank-2 really is 2 dimensional, with a "genweb" running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally. When the period is an octave, this octave-reduces to a single horizontal genchain. But shrutal has a genweb with vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth.<br />
<br />
In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.<br />
<br />
If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Shrutal's generator could be thought of as either 3/2 or 16/15, because 16/15 would still create the same mode numbers and thus the same scale names.<br />
<br />
All five Shrutal [10] modes:<br />
<table class="wiki_table">
<tr>
<td>scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>1st genchain<br />
</td>
<td>2nd genchain<br />
</td>
</tr>
<tr>
<td>1st Shrutal [10]<br />
</td>
<td>ssssL-ssssL<br />
</td>
<td>C C# D D# E F# G G# A A# C<br />
</td>
<td><u><strong>C</strong></u> G D A E<br />
</td>
<td>F# C# G# D# A#<br />
</td>
</tr>
<tr>
<td>2nd Shrutal [10]<br />
</td>
<td>sssLs-sssLs<br />
</td>
<td>C C# D D# F F# G G# A B C<br />
</td>
<td>F <u><strong>C</strong></u> G D A<br />
</td>
<td>B F# C# G# D#<br />
</td>
</tr>
<tr>
<td>3rd Shrutal [10]<br />
</td>
<td>ssLss-ssLss<br />
</td>
<td>C C# D E F F# G G# Bb B C<br />
</td>
<td>Bb F <u><strong>C</strong></u> G D<br />
</td>
<td>E B F# C# G#<br />
</td>
</tr>
<tr>
<td>4th Shrutal [10]<br />
</td>
<td>sLsss-sLsss<br />
</td>
<td>C C# Eb E F F# G A Bb B C<br />
</td>
<td>Eb Bb F <u><strong>C</strong></u> G<br />
</td>
<td>A E B F# C#<br />
</td>
</tr>
<tr>
<td>5th Shrutal [10]<br />
</td>
<td>Lssss-Lssss<br />
</td>
<td>C D Eb E F F# Ab A Bb B C<br />
</td>
<td>Ab Eb Bb F <u><strong>C</strong></u><br />
</td>
<td>D A E B F#<br />
</td>
</tr>
</table>
<br />
The octotonic diminished scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four short genchains.<br />
<table class="wiki_table">
<tr>
<td>scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>1st chain<br />
</td>
<td>2nd chain<br />
</td>
<td>3rd chain<br />
</td>
<td>4th chain<br />
</td>
</tr>
<tr>
<td>1st Diminished[8]<br />
</td>
<td>sLsL sLsL<br />
</td>
<td>C Db Eb E Gb G A Bb C<br />
</td>
<td style="text-align: center;"><u><strong>C</strong></u> G<br />
</td>
<td>Eb Bb<br />
</td>
<td>Gb Db<br />
</td>
<td>A E<br />
</td>
</tr>
<tr>
<td>2nd Diminished[8]<br />
</td>
<td>LsLs LsLs<br />
</td>
<td>C D Eb F F# Ab A B C<br />
</td>
<td style="text-align: center;">F <u><strong>C</strong></u><br />
</td>
<td>Ab Eb<br />
</td>
<td>B F#<br />
</td>
<td>D A<br />
</td>
</tr>
</table>
<br />
There are only two Blackwood[10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. Ups and downs are used to distinguish between 5/4 and 2\5, in order to avoid duplicate note names.<br />
<table class="wiki_table">
<tr>
<td>scale name<br />
</td>
<td>Ls pattern<br />
</td>
<td>example in C<br />
</td>
<td>1st chain<br />
</td>
<td>2nd chain<br />
</td>
<td>3rd chain<br />
</td>
<td>4th chain<br />
</td>
<td>5th chain<br />
</td>
</tr>
<tr>
<td>1st Blackwood[10]<br />
</td>
<td>LsLsLs LsLs<br />
</td>
<td>C C#v D Ev F F#v G Av A Bv C<br />
</td>
<td style="text-align: center;"><u><strong>C</strong></u> Ev<br />
</td>
<td>D F#v<br />
</td>
<td>F Av<br />
</td>
<td>G Bv<br />
</td>
<td>A C#v<br />
</td>
</tr>
<tr>
<td>2nd Blackwood[10]<br />
</td>
<td>sLsLsL sLsL<br />
</td>
<td>C C^ D Eb^ E F^ G Ab^ A Bb^ C<br />
</td>
<td style="text-align: center;">Ab^ <u><strong>C</strong></u><br />
</td>
<td>Bb^ D<br />
</td>
<td>C^ E<br />
</td>
<td>Eb^ G<br />
</td>
<td>F^ A<br />
</td>
</tr>
</table>
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-Non-MOS non-MODMOS scales"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:22:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales" title="Anchor: How to name rank-2 scales-Non-MOS scales"/> --><a name="How to name rank-2 scales-Non-MOS scales"></a><!-- ws:end:WikiTextAnchorRule:22 --><strong><u>Non-MOS non-MODMOS scales</u></strong></h2>
Compact the genchain to remove any gaps via alterations. The mode number is derived from the compacted genchain. Examples:<br />
<br />
C D E F F# G A B C has the genchain F <u><strong>C</strong></u> G D A E B F#, and is named C 2nd Meantone[8].<br />
<br />
C D E F F# G A Bb C, with genchain Bb F <u><strong>C</strong></u> G D A E * F#. Alter Bb to get an unbroken genchain: F <u><strong>C</strong></u> G D A E B F#. The scale is C 2nd Meantone[8,-7].<br />
<br />
A B C D D# E F G G# A, with genchain F C G D <u><strong>A</strong></u> E B * * G# D#. Sharpen F and C to get an unbroken genchain: G D <u><strong>A</strong></u> E B F# C# G# D#, giving the name A 3rd Meantone[9,-6,-7].<br />
<br />
F G A C E F, with genchain <u><strong>F</strong></u> C G * A E. No amount of altering will make an unbroken genchain, so the name is F 1st Meantone[6,no4].<br />
<br />
A B C E F A, a japanese pentatonic scale, with genchain F C * * <u><strong>A</strong></u> E B. The F and C can't be sharpened to F# and C# to make an unbroken Meantone[5] genchain, because pentatonic alterations move notes by 5 steps, not 7. No amount of <u>pentatonic</u> altering will make an unbroken genchain, so the scale must be named as a heptatonic scale with missing degrees: A 5th Meantone[7,no3,no4].<br />
<br />
<em>[Problem: The 2nd and 3rd examples use heptatonic alterations even though the scales aren't heptatonic.]</em><br />
<em>[Problem: octotonic alterations would be absurd!]</em><br />
<em>[Possible solution: non-MOS scales can't have chromatic alterations, only missing notes. 2nd example is 3rd Meantone[9,no8]. 3rd example is 5th Meantone[11,no8,no9].</em><br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-Explanation / Rationale"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:23:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@How to name rank-2 scales-Non-MOS scales" title="Anchor: How to name rank-2 scales-Non-MOS scales"/> --><a name="How to name rank-2 scales-Non-MOS scales"></a><!-- ws:end:WikiTextAnchorRule:23 --><u>Explanation / Rationale</u></h2>
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x-Explanation / Rationale-Why not number the modes in the order they occur in the scale?"></a><!-- ws:end:WikiTextHeadingRule:10 --><strong><u>Why not number the modes in the order they occur in the scale?</u></strong></h3>
<br />
Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.<br />
<br />
<u>Genchain-based</u>: if the Meantone[7] genchain were notated 1 2 3 4 5 6 7, the Lydian scale would be 1 3 5 7 2 4 6 1, and the major scale would be 2 4 6 1 3 5 7 2.<br />
<br />
<u>Scale-based</u>: if the Meantone[7] major scale were notated 1 2 3 4 5 6 7 1, the genchain would be 4 1 5 2 6 3 7.<br />
<br />
The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The modes are ordered in a spectrum, and the 1st and last modes are always the two most extreme. For MOS scales, adjacent modes differ by only one note.<br />
<br />
The disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="x-Explanation / Rationale-Why make an exception for 3/2 vs 4/3 as the generator?"></a><!-- ws:end:WikiTextHeadingRule:12 --><u><strong>Why make an exception for 3/2 vs 4/3 as the generator?</strong></u></h3>
<br />
Because of centuries of established thought that the fifth, not the fourth, generates the pythagorean, meantone and well tempered scales, as these quotes show:<br />
<br />
"Pythagorean tuning is a tuning of the syntonic temperament in which the <span class="mw-redirect">generator</span> is the ratio <u><strong><span class="mw-redirect">3:2</span></strong></u> (i.e., the untempered perfect <u><strong>fifth</strong></u>)." -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Pythagorean_tuning" rel="nofollow">en.wikipedia.org/wiki/Pythagorean_tuning</a><br />
<br />
"The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect <u><strong>fifth</strong></u>." -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Syntonic_temperament" rel="nofollow">en.wikipedia.org/wiki/Syntonic_temperament</a><br />
<br />
"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect <u><strong>fifths</strong></u>." --<br />
<a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Meantone_temperament" rel="nofollow">en.wikipedia.org/wiki/Meantone_temperament</a><br />
<br />
"In this system the perfect <u><strong>fifth</strong></u> is flattened by one quarter of a syntonic comma." -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Quarter-comma_meantone" rel="nofollow">en.wikipedia.org/wiki/Quarter-comma_meantone</a><br />
<br />
"The term "well temperament" or "good temperament" usually means some sort of <span class="new">irregular temperament</span> in which the tempered <u><strong>fifths</strong></u> are of different sizes." -- <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Well_temperament" rel="nofollow">en.wikipedia.org/wiki/Well_temperament</a><br />
<br />
"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike <u>wise</u> consistencies, it wouldn't reduce memorization, because everyone already knows that the generator is historically 3/2.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="x-Explanation / Rationale-Then why not always choose the larger of the two generators?"></a><!-- ws:end:WikiTextHeadingRule:14 --><u><strong>Then why not always choose the larger of the two generators?</strong></u></h3>
<br />
Because the interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x-Explanation / Rationale-Why not always choose the chroma-positive generator?"></a><!-- ws:end:WikiTextHeadingRule:16 --><u>Why not always choose the chroma-positive generator?</u></h3>
<br />
See below.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x-Explanation / Rationale-Why not just use UDP notation?"></a><!-- ws:end:WikiTextHeadingRule:18 --><u><strong>Why not just use UDP notation?</strong></u></h3>
<br />
One problem with UDP is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction.<br />
<table class="wiki_table">
<tr>
<td>scale<br />
</td>
<td>UDP generator<br />
</td>
<td>UDP genchain<br />
</td>
<td>Mode Numbers generator<br />
</td>
<td>Mode Numbers genchain<br />
</td>
</tr>
<tr>
<td>Meantone[5] in 31edo<br />
</td>
<td style="text-align: center;">4/3<br />
</td>
<td>E A D G C<br />
</td>
<td style="text-align: center;">3/2<br />
</td>
<td>C G D A E<br />
</td>
</tr>
<tr>
<td>Meantone[7] in 31edo<br />
</td>
<td style="text-align: center;">3/2<br />
</td>
<td>C G D A E B F#<br />
</td>
<td style="text-align: center;">3/2<br />
</td>
<td>C G D A E B F#<br />
</td>
</tr>
<tr>
<td>Meantone[12] in 31edo<br />
</td>
<td style="text-align: center;">4/3<br />
</td>
<td>E# A# D# G# C# F# B E<br />
A D G C<br />
</td>
<td style="text-align: center;">3/2<br />
</td>
<td>C G D A E B F# C# G#<br />
D# A# E#<br />
</td>
</tr>
<tr>
<td>Meantone[19] in 31edo<br />
</td>
<td style="text-align: center;">3/2<br />
</td>
<td>C G D A E B F# C# G#<br />
D# A# E# B# Fx Cx Gx<br />
Dx Ax Ex<br />
</td>
<td style="text-align: center;">3/2<br />
</td>
<td>C G D A E B F# C# G#<br />
D# A# E# B# Fx Cx Gx<br />
Dx Ax Ex<br />
</td>
</tr>
</table>
<br />
A larger problem is that choosing the chroma-positive generator only applies to MOS and MODMOS scales, and breaks down when the length of the genchain results in a non-MOS scale. Mode Numbers notation can be applied to scales like Meantone[8], which while not a MOS, is certainly musically useful.<br />
<table class="wiki_table">
<tr>
<td>scale<br />
</td>
<td>UDP genchain<br />
</td>
<td>Mode Numbers genchain<br />
</td>
</tr>
<tr>
<td>Meantone [2]<br />
</td>
<td>C G<br />
</td>
<td>C G<br />
</td>
</tr>
<tr>
<td>Meantone [3]<br />
</td>
<td>D G C<br />
</td>
<td>C G D<br />
</td>
</tr>
<tr>
<td>Meantone [4]<br />
</td>
<td>???<br />
</td>
<td>C G D A<br />
</td>
</tr>
<tr>
<td>Meantone [5]<br />
</td>
<td>E A D G C<br />
</td>
<td>C G D A E<br />
</td>
</tr>
<tr>
<td>Meantone [6]<br />
</td>
<td>???<br />
</td>
<td>G C D A E B<br />
</td>
</tr>
<tr>
<td>Meantone [7]<br />
</td>
<td>C G D A E B F#<br />
</td>
<td>C G D A E B F#<br />
</td>
</tr>
<tr>
<td>Meantone [8]<br />
</td>
<td>???<br />
</td>
<td>C G D A E B F# C#<br />
</td>
</tr>
<tr>
<td>Meantone [9]<br />
</td>
<td>???<br />
</td>
<td>C G D A E B F# C# G#<br />
</td>
</tr>
<tr>
<td>Meantone [10]<br />
</td>
<td>???<br />
</td>
<td>C G D A E B F# C# G# D#<br />
</td>
</tr>
<tr>
<td>Meantone [11]<br />
</td>
<td>???<br />
</td>
<td>C G D A E B F# C# G# D# A#<br />
</td>
</tr>
<tr>
<td>Meantone [12] if generator < 700¢<br />
</td>
<td>E# A# D# G# C# F# B E A D G C<br />
</td>
<td>C G D A E B F# C# G# D# A# E#<br />
</td>
</tr>
<tr>
<td style="text-align: left;">Meantone [12] if generator > 700¢<br />
</td>
<td>C G D A E B F# C# G# D# A# E#<br />
</td>
<td style="text-align: center;">C G D A E B F# C# G# D# A# E#<br />
</td>
</tr>
</table>
<br />
An even larger problem is that the notation is overly tuning-dependent. Meantone[12] generated by 701¢ has a different genchain than Meantone[12] generated by 699¢, so slight differences in tempering result in different mode names. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth[12]'s fifth to be more than 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either more or less than 700¢. This makes every single UDP mode of Dominant[12] ambiguous. For example "Dominant 8|3" could mean either "4th Dominant[12]" or "9th Dominant[12]". Something similar happens with Meantone[19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone[19] ambiguous. Another example is Dicot[7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore[5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.<br />
<br />
A fourth problem with UDP is more of a taste issue: UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented. For example, the most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). Also, as noted above, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. Furthermore, UDP uses the more mathematical <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Zero-based_numbering" rel="nofollow">zero-based counting</a> and Mode Numbers notation uses the more intuitive one-based counting.<br />
<br />
<br />
<br />
<br />
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