Xenharmonic Wiki - User contributions [en]

User:Francium/647edo

2026-05-13T00:17:10Z

Nick Vuci: Nominated for deletion due to violating notability guidelines

{{delete|[[notability guidelines|violates notability guidelines]]}}{{Infobox ET}}
{{ED intro}}

== Theory ==
647edo is [[consistent]] to the [[7-odd-limit]] and its [[harmonic]] [[3/1|3]] is about halfway its steps. It tends heavily flat in the first few harmonics. Using the [[patent val]] it [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032 and 1224440064/1220703125 in the [[7-limit]]; [[8019/8000]], 2401/2400, 46656/46585 and 1265625/1261568 in the [[11-limit]]. Alternatively, the 2.9.5.7.11.13 [[subgroup]] might be worth considering.

=== Odd harmonics ===
{{Harmonics in equal|647}}

=== Subsets and supersets ===
647edo is the 118th [[prime EDO]]. [[1294edo]], which doubles it, gives a good correction to the harmonic 3.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.9
| {{monzo|2051 -647}}
| {{mapping|647 2051}}
| −0.0171
| 0.0171
| 0.92
|-
| 2.9.5
| {{monzo|-50 -4 27}}, {{monzo|63 -25 7}}
| {{mapping|647 2051 1502}}
| +0.0651
| 0.1172
| 6.32
|-
| 2.9.5.7
| 703125/702464, 3500000000/3486784401, 4202539929/4194304000
| {{mapping|647 2051 1502 1816}}
| +0.1081
| 0.1258
| 6.78
|-
| 2.9.5.7.11
| 496125/495616, 1684375/1679616, 151263/151250, 40960000/40920957
| {{mapping|647 2051 1502 1816 2238}}
| +0.1135
| 0.1131
| 6.10
|-
| 2.9.5.7.11.13
| 4459/4455, 35750/35721, 47432/47385, 60025/59904, 1146880/1146717
| {{mapping|647 2051 1502 1816 2238 2394}}
| +0.1100
| 0.1035
| 5.58
|}

== Music ==
; [[Francium]]
* "Freeware" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/2OFMrhE9KtqfCj3RhPjFjp Spotify] | [https://francium223.bandcamp.com/track/freeware Bandcamp] | [https://www.youtube.com/watch?v=QSiZsJjnu78 YouTube]

Talk:Frequency ratio

2025-11-25T16:48:44Z

Nick Vuci: Created page with "== Extended Frequency Ratio Section and Table of Contents == The recent additions are way too much for the page. It went from simple and clear to cluttered and confusing, which is not appropriate for a beginner page. I recommend removing the table of contents and the Extended Frequency Ratio section entirely as it touches upon more advanced topics that are discussed more clearly elsewhere, and the body has now been updated to include mention that ratios with more than tw..."

== Extended Frequency Ratio Section and Table of Contents ==
The recent additions are way too much for the page. It went from simple and clear to cluttered and confusing, which is not appropriate for a beginner page. I recommend removing the table of contents and the Extended Frequency Ratio section entirely as it touches upon more advanced topics that are discussed more clearly elsewhere, and the body has now been updated to include mention that ratios with more than two terms can be specified as extended ratios if necessary. This simple fact shouldn't take up half the page or be presented as something entirely different than ratios with two terms.

Furthermore, the statement that these are called "EFRs" or that subharmonic series chords are called "SEFRs" is incorrect. This term was made up in recent times simply because the actual term wasn't known, so presenting these as things beginners should learn or as things people commonly use is inappropriate.

--[[User:Nick Vuci|Nick Vuci]] ([[User talk:Nick Vuci|talk]]) 16:48, 25 November 2025 (UTC)

Frequency ratio

2025-11-25T16:28:46Z

Nick Vuci: Added details about conventions for writing ratios with more than two terms in main body.

{{interwiki
| en = Frequency ratio
| de =
| es = Radio
| ja = 周波数比率
}}
A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).

All [[interval]]s can be expressed as ratios, and they can be rational or irrational. Although mostly written in the form ''larger''/''smaller'' throughout this wiki, they may be written in several ways:
* 2/1, 2:1, 1/2, 1:2 for the [[octave]];
* 3/2, 3:2, 2/3, 2:3 for the [[3/2|just perfect fifth]].

When the larger number is written first (''note''/''base''), this usually signifies a note being played ''above'' some base tone (perhaps the starting note of a scale). When the smaller number is written first (''base''/''note''), this usually signifies the note being played ''below'' that base tone.

Ratios with more than two terms (sometimes called '''extended ratios''') can be used to express [[chord]]s and [[Scale|scales]]. For example, the just intoned major triad in root position is 4:5:6. Chords can also be written as a string of intervals, such as 1/1–5/4–3/2. (4:5:6 can be viewed as a shorthand for 4/1:5/1:6/1 or 4/4:5/4:6/4).

The [[harmonic series]] can be represented as the infinite ratio 1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:16:17… Segments of the harmonic series are commonly written in abbreviated form with a double colon. For example, 8:9:10:11:12:13:14:15:16 is commonly written as 8::16.

In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".

== Conversion ==
{{See also| Cent #Conversion | Monzo #Examples }}

=== Monzo to ratio ===
To find the ratio ''r'' for an interval of monzo '''m''' = {{monzo| ''m''1 ''m''2 ''m''3 … }}, apply

<math>\displaystyle r = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \cdot \ldots</math>

=== Cents to ratio ===
To find the ratio ''r'' for an interval of ''s'' cents, apply

<math>\displaystyle r = 2^{s/1200}</math>

== Extended frequency ratio (EFR) ==
An extended ratio is a ratio of more than 2 numbers. For example, a recipe might combine cups of flour, milk and sugar in a 5:3:4 ratio. Xenharmonic music theory uses an extended ratio to indicate the various frequency ratios between three or more notes. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous unordered example, EFRs are ordered, with harmonic EFR typically written in an ascending order and subharmonic EFR typically written in a descending order.

=== Harmonic EFR ===
For example, consider a [[just intonation]] major triad on {{w|A440}} with a ratio list of 1/1–5/4–3/2. The three frequencies are 440, 550 and 660. The EFR is 440:550:660, which simplifies to [[4:5:6]], spoken as "four five six". Had the root been other than A440, the EFR would be the same.

To convert an EFR to a ratio list, simply divide every harmonic by the first harmonic. For example, (4:5:6)/4 is 4/4–5/4–6/4, which simplifies to 1/1–5/4–3/2.

The EFR directly indicates the interval between any pair of notes in the chord. In the harmonic seventh chord [[4:5:6:7]], the interval between the third and the fifth is 6/5, that between the third and seventh is 7/5, etc.

The EFR also indicates where in the harmonic series the chord occurs. 4:5:6:7 occurs as [[harmonic]]s 4, 5, 6 and 7. Thus an EFR is also a list of harmonics.

To convert a list of ratios to a list of harmonics, multiply each ratio by the LCM of the denominators. For example, 1/1–6/5–3/2 has denominators 1, 5 and 2, with an LCM of 10. Multiplying each ratio by 10 makes 10/1–12/1–15/1. Remove the ones to get 10:12:15.

=== Subharmonic EFR, a.k.a. SEFR ===
Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Some writers omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as /7:6:5:4 or 7:6:5:4.

To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.

The SEFR directly indicates the interval between any pair of notes in the chord. In 1/(7:6:5:4), the interval between the third and the fifth is 6/5, that between the third and seventh is 6/4 which is 3/2, etc.

To convert a list of ratios to a list of subharmonics, divide each ratio by the LCM of the numerators. For example, 1/1–6/5–3/2 has numerators 1, 6 and 3, with an LCM of 6. Dividing each ratio by 6 makes 1/6–1/5–1/4. Extract the ones to get 1/(6:5:4).

To convert between EFR from and to SEFR, do the following:

* [to SEFR] 10:12:15 (write as reciprocals) → 1/(1/10:1/12:1/15) → (multiply by LCM and reduce) → 60/(1/10:1/12:1/15) = 1/(6:5:4)
* [to EFR] 1/(6:5:4) (write as reciprocals) → 1/6:1/5:1/4 → (multiply by LCM and reduce) → 60*(1/6:1/5:1/4) = 10:12:15

Any SEFR can be written as a EFR and viceversa. In theory, 1/(7:6:5:4) could be written as [[60:70:84:105]], and so could 4:5:6:7 be written as 1/(105:84:70:60). The first option is more common, as seen in the available link. Ultimately, the choice depends on clarity.

Some chords are [[Self-invertible chord|self-inverses]], making SEFRs unnecessary. Examples include [[8:10:12:15]] (pental major seventh), [[10:12:15:18]] (pental minor seventh), [[12:14:18:21]] (septimal minor seventh), and [[14:18:21:27]] (septimal major seventh). These chords often have [[Homonym|homonyms]], and their inversions may simply be other rotations of the same set. For example, 9:10:12:15 (pental major sixth, 2nd inversion) inverts to 12:15:18:20 (idem, root position), both rotations of 10:12:15:18.

=== Alternative forms ===
Both ratio lists and EFRs can indicate the voicing of a chord. For example, a 4:5:6 major triad in [[Kite's thoughts on hi-lo notation|hi3add8 voicing]] is 1/1–3/2–2/1–5/2 or 2:3:4:5.

Contiguous harmonics such as ''n'':(''n'' + 1):(''n'' + 2):(''n'' + 3) can be written with a double colon as ''n''::(''n'' + 3). Likewise for contiguous subharmonics. This is especially common for scales like 8::16.

[[Category:Ratio| ]] 
[[Category:Notation]]
[[Category:Terms]]

User:Nick Vuci/Moments of Symmetry

2025-10-27T00:36:16Z

Nick Vuci: /* Spectrum of MOS */

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="slideshow" widths="200" heights="200">
File:MOS Spectrum.gif|The entire MOS spectrum at 6-levels
File:MOS Spectrum mirror.gif|Showing the mirror-like symmetry of the half period. The same patterns repeat from 0-600 cents and 1200-600 cents.
File:MOS Spectrum Level2.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

File:MOS Spectrum mirror.gif

2025-10-27T00:34:53Z

Nick Vuci: MOS spectrum showing the mirror symmetry of the half-period mark

== Summary ==
MOS spectrum showing the mirror symmetry of the half-period mark
== Licensing ==
{{self|cc-by-sa-4.0}}

User:Nick Vuci/Moments of Symmetry

2025-10-27T00:32:05Z

Nick Vuci: /* Spectrum of MOS */

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="slideshow" widths="200" heights="200">
File:MOS Spectrum.gif|The entire MOS spectrum at 6-levels
File:MOS Spectrum Level2.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T17:54:19Z

Nick Vuci: /* Spectrum of MOS */

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="slideshow" widths="200" heights="200">
File:MOS Spectrum.gif
File:MOS Spectrum Level2.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T17:53:54Z

Nick Vuci: /* Step Ratios */

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="nolines" widths="200" heights="200">
File:MOS Spectrum.gif
File:MOS Spectrum Level2.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T17:36:41Z

Nick Vuci: /* Spectrum of MOS */

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="nolines" widths="300" heights="300">
File:MOS Spectrum.gif
File:MOS Spectrum Level2.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T17:36:10Z

Nick Vuci:

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="nolines" widths="300" heights="300">
File:MOS Spectrum.gif
File:MOS Spectrum Level1.gif
File:MOS Spectrum Level2.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T17:30:24Z

Nick Vuci: added spectrum gifs

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="nolines">
File:MOS Spectrum.gif|The complete MOS spectrum for the 6-level horogram.
File:MOS Spectrum Level3.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

File:MOS Spectrum Level3.gif

2025-10-25T17:28:57Z

Nick Vuci: GIF of the MOS spectrum in the third level

== Summary ==
GIF of the MOS spectrum in the third level
== Licensing ==
{{self|cc-by-sa-4.0}}

File:MOS Spectrum Level2.gif

2025-10-25T17:28:20Z

Nick Vuci: GIF of the MOS spectrum in the second level

== Summary ==
GIF of the MOS spectrum in the second level
== Licensing ==
{{self|cc-by-sa-4.0}}

File:MOS Spectrum Level1.gif

2025-10-25T17:27:09Z

Nick Vuci: GIF of the MOS spectrum in the first level

== Summary ==
GIF of the MOS spectrum in the first level
== Licensing ==
{{self|cc-by-sa-4.0}}

User:Nick Vuci/Moments of Symmetry

2025-10-25T17:24:53Z

Nick Vuci:

WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery>
File:MOS Spectrum.gif|The complete MOS spectrum for the 6-level horogram.
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T06:37:13Z

Nick Vuci: /* Spectrum of MOS */

WORK-IN-PROGRESS AS OF 27 MAY 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
[[File:MOS Spectrum.gif|thumb|The complete MOS spectrum for a 6-level horogram.]]
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T06:36:45Z

Nick Vuci: /* Spectrum of MOS */

WORK-IN-PROGRESS AS OF 27 MAY 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
[[File:MOS Spectrum.gif|thumb|The complete MOS spectrum for a 6-level horogram.]]
As a generator moves through all possible positions within a fixed period, MOS emerge as specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

User:Nick Vuci/Moments of Symmetry

2025-10-25T06:06:02Z

Nick Vuci: /* Construction */

WORK-IN-PROGRESS AS OF 27 MAY 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]].

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated:

As a concrete step pattern: 2 2 1 2 2 2 1

As an abstract step pattern: L L s L L L s

As a scale signature: 5L 2s

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with 2 step sizes. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with 2 step sizes. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with 2 step sizes. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern.

For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
[[File:MOS Spectrum.gif|thumb]]
The '''MOS spectrum''' is the sequence of MOS that appear as a generator moves through all possible positions within a fixed period. Each point in this sweep defines a generator–period pair that yields an MOS, producing a full map of where two-step structures occur.

=== Core properties ===

* The spectrum is bounded by the equal divisions that mark the range of distinct L/s pattern formation.
* It is symmetric (mirrored) around the midpoint of the period, because generators larger than half the period produce the same MOS as their complements (period − generator).

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]]
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==

File:MOS Spectrum.gif

2025-10-25T04:43:11Z

Nick Vuci:

A gif of the complete spectrum of MOS in 6 levels

Horogram

2025-10-25T02:00:27Z

Nick Vuci: /* Examples */ changed gallery format

A horogram is a diagram that visualizes the [[interval]] pattern for a [[scale]] based on a repeated [[Generator|generating]] interval. This is useful for visualizing the step pattern for [[MOS|moment-of-symmetry scales]] and other similar scales.

Horograms are primarily the work of [[Erv Wilson]].

== Diagrams ==

=== Circular horogram ===
The most familiar form of the horogram is based on Wilson's work. Here, the size of the generating intervals are represented as angles, and successive circles radiating from the center represent increasingly chromatic (fine) child scales.

==== Examples ====
<gallery mode="nolines">
File:5EDO Horogram P1200G720L3.png|720 cent generator ending in 5edo
File:7EDO-Horogram P1200G685.71L4.png|685.71 cent generator ending in 7edo
File:9EDO Horogram P1200G666.67L5.png|666.67 cent generator ending in 9edo
File:10EDO Horogram P1200G840L5.png|840 cent generator ending in 10edo
File:11EDO Horogram P1200G763.64L5.png|763.64 cent generator ending in 11edo
File:13EDO Horogram P1200G738.46L5.png|738.46 cent generator ending in 13edo
File:14EDO Horogram P1200G771.43L6.png|771.43 cent generator ending in 14edo
File:15EDO Horogram P1200G880L6.png|880 cent generator ending in 15edo
</gallery>
==== Construction ====
The construction of a circular horogram is based on rotating a pointer hand by the same angle repeatedly.

{{todo|inline=1|complete section|text=Explain in step by step detail.}}

=== Rectangular horogram ===
A rectangular version of a horogram conveys the same information as a circular horogram but in a rectangular format.

==== Example ====
The use of a spreadsheet application, such as Microsoft Excel, makes it easy to construct a rectangular horogram, as well as representing the information as a table. Shown below is the rectangular horogram for [[12edo]] diatonic ([[5L 2s]]).
{| class="wikitable"
! colspan="12" |Steps for Generators 7\12 and 5\12
!Mos
!Step Ratio
|-
| colspan="7" |7
| colspan="5" |5
|1L 1s
|7:5
|-
| colspan="2" |2
| colspan="5" |5
| colspan="5" |5
|2L 1s
|5:2
|-
| colspan="2" |2
| colspan="2" |2
| colspan="3" |3
| colspan="2" |2
| colspan="3" |3
|2L 3s
|3:2
|-
| colspan="2" |2
| colspan="2" |2
| colspan="2" |2
|1
| colspan="2" |2
| colspan="2" |2
|1
|5L 2s
|2:1
|-
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|12edo
|1
|}

==== Construction ====
One way to conceptualize the construction of a rectangular horogram is to think of it as a partitioning a rectangle until every partition is of the same size. The algorithm for partitioning is shown below for a rectangle of length e initially partitioned into two such that one of the partitions is of length g.

# For a rectangle of length e, partition a section of length g from the right of that rectangle. This splits the rectangle into two parts of sizes (e - g)\e and g\e, from left to right.
# Assign the larger of the two partitions as partition L and the smaller of the two as s.
# If, from left to right, the larger of the two partitions precedes the smaller one, the larger partition is partitioned such that a partition of size s is broken off from the right. Otherwise, if the smaller partition precedes the larger one, a partition of size s is partitioned from the left.
# In either case, update the running values of L and s and repeat the previous step. If partitions L and s are of the same size, then no further partitioning is possible.

== Applications ==

=== Moment-of-symmetry scales ===
{{todo|inline=1|complete section}}

=== Max-variety-3 scales ===
{{todo|inline=1|complete section}}

== External links ==
* [http://anaphoria.com/wilson.html The Wilson Archives] (contains various PDF files depicting horograms)

{{todo|inline=1|add examples}}

[[Category:Horogram]]
[[Category:Erv Wilson]]

Horogram

2025-10-25T01:59:04Z

Nick Vuci: /* Example */ Added circular horogram examples

A horogram is a diagram that visualizes the [[interval]] pattern for a [[scale]] based on a repeated [[Generator|generating]] interval. This is useful for visualizing the step pattern for [[MOS|moment-of-symmetry scales]] and other similar scales.

Horograms are primarily the work of [[Erv Wilson]].

== Diagrams ==

=== Circular horogram ===
The most familiar form of the horogram is based on Wilson's work. Here, the size of the generating intervals are represented as angles, and successive circles radiating from the center represent increasingly chromatic (fine) child scales.

==== Examples ====
<gallery mode="packed-overlay">
File:5EDO Horogram P1200G720L3.png|720 cent generator ending in 5edo
File:7EDO-Horogram P1200G685.71L4.png|685.71 cent generator ending in 7edo
File:9EDO Horogram P1200G666.67L5.png|666.67 cent generator ending in 9edo
File:10EDO Horogram P1200G840L5.png|840 cent generator ending in 10edo
File:11EDO Horogram P1200G763.64L5.png|763.64 cent generator ending in 11edo
File:13EDO Horogram P1200G738.46L5.png|738.46 cent generator ending in 13edo
File:14EDO Horogram P1200G771.43L6.png|771.43 cent generator ending in 14edo
File:15EDO Horogram P1200G880L6.png|880 cent generator ending in 15edo
</gallery>
==== Construction ====
The construction of a circular horogram is based on rotating a pointer hand by the same angle repeatedly.

{{todo|inline=1|complete section|text=Explain in step by step detail.}}

=== Rectangular horogram ===
A rectangular version of a horogram conveys the same information as a circular horogram but in a rectangular format.

==== Example ====
The use of a spreadsheet application, such as Microsoft Excel, makes it easy to construct a rectangular horogram, as well as representing the information as a table. Shown below is the rectangular horogram for [[12edo]] diatonic ([[5L 2s]]).
{| class="wikitable"
! colspan="12" |Steps for Generators 7\12 and 5\12
!Mos
!Step Ratio
|-
| colspan="7" |7
| colspan="5" |5
|1L 1s
|7:5
|-
| colspan="2" |2
| colspan="5" |5
| colspan="5" |5
|2L 1s
|5:2
|-
| colspan="2" |2
| colspan="2" |2
| colspan="3" |3
| colspan="2" |2
| colspan="3" |3
|2L 3s
|3:2
|-
| colspan="2" |2
| colspan="2" |2
| colspan="2" |2
|1
| colspan="2" |2
| colspan="2" |2
|1
|5L 2s
|2:1
|-
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|1
|12edo
|1
|}

==== Construction ====
One way to conceptualize the construction of a rectangular horogram is to think of it as a partitioning a rectangle until every partition is of the same size. The algorithm for partitioning is shown below for a rectangle of length e initially partitioned into two such that one of the partitions is of length g.

# For a rectangle of length e, partition a section of length g from the right of that rectangle. This splits the rectangle into two parts of sizes (e - g)\e and g\e, from left to right.
# Assign the larger of the two partitions as partition L and the smaller of the two as s.
# If, from left to right, the larger of the two partitions precedes the smaller one, the larger partition is partitioned such that a partition of size s is broken off from the right. Otherwise, if the smaller partition precedes the larger one, a partition of size s is partitioned from the left.
# In either case, update the running values of L and s and repeat the previous step. If partitions L and s are of the same size, then no further partitioning is possible.

== Applications ==

=== Moment-of-symmetry scales ===
{{todo|inline=1|complete section}}

=== Max-variety-3 scales ===
{{todo|inline=1|complete section}}

== External links ==
* [http://anaphoria.com/wilson.html The Wilson Archives] (contains various PDF files depicting horograms)

{{todo|inline=1|add examples}}

[[Category:Horogram]]
[[Category:Erv Wilson]]

File:15EDO Horogram P1200G880L6.png

2025-10-25T01:53:50Z

Nick Vuci: Horogram with a 880 cent generator ending in 15edo

== Summary ==
Horogram with a 880 cent generator ending in 15edo
== Licensing ==
{{self|cc-by-sa-4.0}}

File:14EDO Horogram P1200G771.43L6.png

2025-10-25T01:49:24Z

Nick Vuci: Horogram with a 771.43 cent generator ending in 14edo

== Summary ==
Horogram with a 771.43 cent generator ending in 14edo
== Licensing ==
{{self|cc-by-sa-4.0}}

File:13EDO Horogram P1200G738.46L5.png

2025-10-25T01:46:36Z

Nick Vuci: Horogram with a 738.46 cent generator ending in 13edo

== Summary ==
Horogram with a 738.46 cent generator ending in 13edo
== Licensing ==
{{self|cc-by-sa-4.0}}

File:11EDO Horogram P1200G763.64L5.png

2025-10-25T01:45:12Z

Nick Vuci: Horogram with a 763.64 cent generator ending in 11edo

== Summary ==
Horogram with a 763.64 cent generator ending in 11edo
== Licensing ==
{{self|cc-by-sa-4.0}}

File:10EDO Horogram P1200G840L5.png

2025-10-25T01:42:50Z

Nick Vuci: Horogram with an 840 cent generator ending in 10edo

== Summary ==
Horogram with an 840 cent generator ending in 10edo
== Licensing ==
{{self|cc-by-sa-4.0}}

File:9EDO Horogram P1200G666.67L5.png

2025-10-25T01:41:25Z

Nick Vuci: Horogram with a 666.67 cent generator ending in 9edo

== Summary ==
Horogram with a 666.67 cent generator ending in 9edo
== Licensing ==
{{self|cc-by-sa-4.0}}

File:7EDO-Horogram P1200G685.71L4.png

2025-10-25T01:35:47Z

Nick Vuci:

Horogram with a 685.71 cent generator ending in 7edo

File:5EDO Horogram P1200G720L3.png

2025-10-25T01:32:03Z

Nick Vuci:

Image of a horogram with 3 levels and a 720 cent generator resulting in 5edo

Xenharmonic Wiki:Elections

2025-10-21T23:42:36Z

Nick Vuci: /* Support */

'''Elections''' is where the community decides decides who will become [[administrators]] or [[bureaucrats]].

To vote, simply put your comments and signature under the relevant section:

<code><nowiki># '''Support'''. I think this is a very good candidate! ~~~~</nowiki></code>

You can also add comments or questions for the candidate under 'comments'.

==Fredg999==

===Nomination===

I would like to nominate [[User:Fredg999]] for [[Xenharmonic_Wiki:Bureaucrats|bureaucrat]] (he is already an admin currently). The wiki currently has very few active bureaucrats and I think it's a good idea to have at least one more.

I'll keep it brief. Fredg999 has a track record of good edits, mostly focused on improving the general quality and usability of the wiki. More importantly though, he has shown himself to be a very level-headed person who is good at conflict resolution.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:45, 21 October 2025 (UTC)

===Support===
# '''Support'''. As a nominator. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:45, 21 October 2025 (UTC)
# '''Support'''. Absolutely. – [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 22:10, 21 October 2025 (UTC)
# '''Support'''. I was hoping someone would propose this actually, I think he'd be great for the role. - [[Budjarn Lambeth]], 22 October 2025 (UTC+10)
# '''Support'''. Can't think of a better person for the job. – [[User:Nick Vuci|Nick Vuci]] ([[User talk:Nick Vuci|talk]]) 23:42, 21 October 2025 (UTC)

===Oppose===

===Neutral===

===Comments===

Talk:Main Page

2025-10-07T22:32:37Z

Nick Vuci: /* Revision proposal by Budjarn ver. 2 (Oct 2025) */

{{Mbox|type=notice|text=<big>'''Welcome! This page is for discussing the contents of the Xenharmonic Wiki's Main Page.'''</big>

For general questions unrelated to the Main Page, please visit [[Help talk:Introduction]].}}
{{Archives}}{{clear}}
{{WSArchiveLink}}

== iOS misidentifying numbers as phone numbers ==
On many pages of the wiki, iOS misidentifies numbers as phone numbers and converts them to links, which can be disorienting because they look indistinguishable from links to other wiki pages. Here is one example:
<gallery>
IOSPhoneBug1.jpeg
IOSPhoneBug2.png
</gallery>

Squarespace allows you to prevent it doing this to your website, by injecting a short line of code: https://sf.digital/squarespace-solutions/how-to-stop-numbers-being-converted-to-links-on-mobile

I was wondering whether MediaWiki has anything similar we could use?

--[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 05:23, 26 January 2024 (UTC)

== Revision proposal by Sintel ==
As the new staff on the wiki, we plan to propound edits to the main page (since much of its content and linkage is rather obsolete at this point) on a basis established by community consensus. This page should be the place to discuss proposed revisions, suggested by admins and non-admins. New proposals will be published on the [[Xenharmonic Wiki:Daily newsfeed|daily newsfeed]], once that page is established, as well.

To that end, Sintel has suggested two preliminary changes:

* Remove the link to Migration FAQ (unnecessary at this point, 7 years after the migration from Wikispaces).
* Remove the link to the Xenrhythm Wiki (no reason specified).

I would like to see discussion about this proposal. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 10:02, 17 February 2025 (UTC)

== Revision proposal by Lériendil ==
Following the creation of the daily newsfeed, I would like to see the newsfeed itself linked on the main page as a source for updates on what's happening on the wiki (and broader community).

I would like to see discussion about this proposal. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 11:11, 17 February 2025 (UTC)

== Revision proposal by Sevish ==
See [[User:Sevish/Welcome to the Xenharmonic Wiki]]. It was posted by [[User:Sevish]] in 2022. --[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 01:25, 18 February 2025 (UTC)

== Revision proposal by fredg999 ==
I think we should mostly follow Sevish's proposal, which includes Sintel's proposal. I'll add some more ideas that could be merged into it:
* I suggest adding internal links on the keywords of the lead section, namely "theory" (to [[General theory]]), "math" (to [[Mathematical theory]]) and "practice" (to [[Practice]]). Since we don't have two separate pages for "musical tuning" and "tuning system", we don't need to add a link on "tuning system".
* I think we should link to [[Xenharmonic Wiki:About]] instead of [[Xenharmonic Wiki:Introduction]] in the lead section. Now that may sound silly, since the former redirects to the latter at the moment, but here's my point: the former should be a page for visitors, while the latter should be a page for users (editors). Actually, the latter might as well be replaced with [[Help:Help]], which in turn we might want to rename to "Help:Introduction" (in line with the corresponding Wikipedia page).
* Seeing as the name of the website is "Xenharmonic Wiki", yet its topic is "musical tuning", we might want to give the readers some more hints to learn about the distinction between key concepts/terms such as "alternative tuning", "microtonal" and "xenharmonic".
* The [[:File:Pts-2-3-5-e2-twtop-tlin.jpg|projective tuning space image]] is a classic, but I'm never sure if we should do something more with it: add a caption, add more than just one image (it might be perceived as too RTT-leaning), etc. A random image of the day might be funny, although we might end up with microtonal accidentals every other day!
* The sentence about the library mentions "electronically-available published works", but many of the links are actually pointing to library websites that help readers identify where they can find the book or document. We might want to remove "electronically-available", since the library covers more than just those.
* Some links need updating, such as [[Useful Tools]] (merged into [[List of music software]]) and [[Microtonal instruments]] (moved to [[Instruments]]).
* In the last section, the words "sign up" should lead to [[Help:How to Get Your Xenwiki Account]].
* We should also keep the last paragraph to a minimum, with only the link to the main help page but without enumerating the stuff that should be listed on that page anyway.

I'm not yet convinced by the idea of the daily newsfeed, but it certainly has potential. I would suggest deploying that at a later time and implement simpler changes first. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 03:23, 18 February 2025 (UTC)

: I agree to all the bulleted changes. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 12:50, 24 February 2025 (UTC)

== Donations link? ==

I'd like to contribute to support the site. Where can I do this? I searched using "contribute", "contribution", "donate", "donation", "subscribe", and "subscription", but did not find anything. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 22:32, 31 December 2024 (UTC)
: (Bumping this on Xenharmonic Wiki:Wikifuture) [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:47, 24 March 2025 (UTC) and again [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 22:30, 10 September 2025 (UTC)

== 2025-04-08 update ==

I updated the Main Page on April 6th and did another (smaller) edit on April 8th, but I did not apply all of the proposed changes yet. Here are the remaining changes:
* Replace link to [[Xenharmonic Wiki:Introduction]] with link to [[Xenharmonic Wiki:About]] in the lead section;
** This will require work on the About page beforehand.
* Add/change/vary images (instead of having only the classic [[PTS]]).
** I haven't progressed on this idea, but I don't want to force it either. It's not critical anyway.

More discussions are ongoing on Discord, namely regarding the difference between the main page and the sidebar, as well as concentrating work on the pages linked on the main page to make sure the links are generally useful. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 04:28, 8 April 2025 (UTC)

== Shouldn't we keep some home page link to Category:People? ==

Seems to me that it would be useful to keep some sort of link to [[People|Category:People]], since some people who contributed to microtonal/xenharmonic music could not be part of the Xenharmonic Alliance for the simple reason of their lives having been in an earlier time period. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 06:39, 8 April 2025 (UTC)

: A category link does not really pass for 'quality content', at least, not enough to be linked directly from main. I think it would be fine to link e.g. from the main [[Microtonal music]] page.
: On that point, the [[Microtonal_music#History|History]] section of that page is a bit of a joke, so I do agree that more should be added there.
: – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:26, 8 April 2025 (UTC)

:: I'll put my vote in for both changes on the Talk page for that. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 19:34, 8 April 2025 (UTC)

== Revision proposal by hkm ==

[[List of xenharmonic music by community ratings]] is ready to be linked on the main page. It is more useful than the Listen page for discovering music already, and it will grow in usefulness faster if more people see it.
: [[User:BudjarnLambeth]] added a link to this page in the lead section of [[Listen]]. Considering the other curated lists that are now available there, I suggest keeping only "Listen" on the main page, in order to avoid giving undue weight to this specific list compared to other existing curated lists. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 05:06, 18 April 2025 (UTC)

== Revision proposal by Budjarn (Sep 2025) ==

Following the result of the poll on the Discord (6 in favour, 0 against) I would like to propose that we change the wording of the first line of the main page:

; Old wording
"The [[Xenharmonic Wiki]] is an open resource for information about [[musical tuning]]. This information includes tuning systems and their relationships, [[theory]], [[math]], and [[practice]]."

; Proposed new wording
"The Xenharmonic Wiki is an open resource for information about [[musical tuning]] and [[microtonal music]], including [[xenharmonic music]] as well as [[historical tunings]] and those of [[world musical traditions]]. This information includes tuning systems and their relationships, [[theory]], [[math]], and [[practice]]."

--[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 13:21, 28 September 2025 (UTC)

: Sounds good to me, except that the link to [[List_of_approaches_to_musical_tuning#Musical_traditions_of_indigenous,_ancient,_and/or_non-Western_cultures|world musical traditions]] doesn't go to the corresponding section. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:04, 28 September 2025 (UTC)

:: Thanks Lucius, I have fixed the link now :) --[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 00:01, 29 September 2025 (UTC)

:: Anybody else have any thoughts on this? You think it's okay to go ahead, or no? --[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 05:10, 30 September 2025 (UTC)

=== Revision proposal by Budjarn ver. 2 (Oct 2025) ===
Because the first proposal failed to gain traction, I decided to reword it a bit and propose it again. The changes are to reflect the fact that, in practice, the wiki is mostly about xenharmonics (so as not to mislead readers who are looking for comprehensive info on world/historical tunings), while still making it very clear that articles about world/historical tunings are still allowed and encouraged.

; Old wording
"The [[Xenharmonic Wiki]] is an open resource for information about [[musical tuning]]. This information includes tuning systems and their relationships, [[theory]], [[math]], and [[practice]]."

; New proposed wording:
"The Xenharmonic Wiki is an open resource for information about [[microtonal music]]: primarily [[xenharmonic music]] but also [[historical tunings]] and those of [[world musical traditions]]. This information includes tuning systems and their relationships, [[theory]], [[math]], and [[practice]]."

Everyone is encouraged to please reply on this talk page to discuss this proposal. Even if you only want to say "I like the proposal" or "I don't like it", that's still a big help :)

--[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 20:36, 7 October 2025 (UTC)

: Looks good. —[[User:Hkm|hkm]] ([[User talk:Hkm|talk]]) 21:13, 7 October 2025 (UTC)

: The elaboration, I strongly agree with. My only complaint is that I would prefer to keep "[[musical tuning]]" over "[[microtonal music]]," as plenty of the tunings discussed on the wiki would be classed as "macrotonal," or even regular-sized-tonal; even 12edo itself is covered here. [[User:Unque|Unque]] ([[User talk:Unque|talk]]) 21:39, 7 October 2025 (UTC)

:: I am happy to change that. So in that case it would be:

:: '''New proposed wording:'''

:: "The Xenharmonic Wiki is an open resource for information about [[musical tuning]]: primarily [[xenharmonic music]] but also [[historical tunings]] and those of [[world musical traditions]]. This information includes tuning systems and their relationships, [[theory]], [[math]], and [[practice]]."

:: --[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 21:51, 7 October 2025 (UTC)

::: I agree with Unque that microtonal is too limiting, but I disagree that "musical tuning" is the appropriate term. "Tuning systems" is more accurate, and also open enough to not exclude anything.
:::-- [[User:Nick Vuci|Nick Vuci]] ([[User talk:Nick Vuci|talk]]) 22:23, 7 October 2025 (UTC)

:::: Happy to do that :)

:::: '''New proposed wording:'''

:::: "The Xenharmonic Wiki is an open resource for information about [[musical tuning|music tuning systems]]: primarily [[xenharmonic music]] but also [[historical tunings]] and those of [[world musical traditions]]. This information includes tuning systems and their relationships, [[theory]], [[math]], and [[practice]]."

:::: --[[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 22:27, 7 October 2025 (UTC)

::::: Not bad, my own suggestion for wording would be:
:::::"The Xenharmonic Wiki is an open resource dedicated to musical tuning systems, focusing on xenharmonic music while also documenting historical tunings and tuning practices from world traditions. It covers the theory and practical applications of these systems."
::::: -- [[User:Nick Vuci|Nick Vuci]] ([[User talk:Nick Vuci|talk]]) 22:32, 7 October 2025 (UTC)

2025-05-28T20:35:45Z

Nick Vuci: